Embed Size (px)
Citation preview
UNIVERSITÉ DE MONTRÉAL
COMPORTEMENT DYNAMIQUE NON LINÉAIRE DES COQUES CYLINDRIQUES
ANISOTROPIQUES NON UNIFORMES SOUMISES À UN
ÉCOULEMENT SUPERSONIQUE
REDOUANE RAMZI
DÉPARTEMENT DE GÉNIE MÉCANIQUE
ÉCOLE POLYTECHNIQUE DE MONTRÉAL
THÈSE PRÉSENTÉE EN VUE DE L’OBTENTION
DU DIPLÔME DE PHILOSOPHIAE DOCTOR
(GÉNIE MÉCANIQUE)
DÉCEMBRE 2012
© Redouane Ramzi, 2012.
UNIVERSITÉ DE MONTRÉAL
ÉCOLE POLYTECHNIQUE DE MONTRÉAL
Cette thèse intitulée :
COMPORTEMENT DYNAMIQUE NON LINÉAIRE DES COQUES CYLINDRIQUES
ANISOTROPIQUES NON UNIFORMES SOUMISES À UN
ÉCOULEMENT SUPERSONIQUE
présentée par : RAMZI Redouane
en vue de l’obtention du diplôme de : Philosophiae Doctor
a été dûment acceptée par le jury d’examen constitué de :
M. BALAZINSKI Marek, Ph.D., président
M. LAKIS Aouni A., Ph.D., membre et directeur de recherche
M. KAHAWITA René, Ph.D., membre
M. NEJAD ENSAN Manouchehr, Ph.D., P.Eng., membre
iii
DÉDICACE
À mes très chers parents
À mes frères et sœurs
iv
REMERCIEMENTS
Je voudrais exprimer mes profondes gratitudes et appréciations à mon directeur de recherche,
Dr. Aouni Lakis, de sa précieuse orientation, de ses encouragements, et conseils sur les attitudes
de recherche tout au long de mes études supérieures, et aussi pour l'occasion qu'il m'a
donnée pour la préparation de mon diplôme de doctorat au sein de l’École Polytechnique.
Je présente mes remerciements spéciaux aux membres du comité du jury de ma thèse,
Dr. Marek Balazinski,
Dr. Aouni A. Lakis,
Dr. René Kahawita,
Dr. Manouchehr Nejad Ensan,
Dr. Gérald J. Zagury,
pour le temps consacré à l’évaluation de mon travail de recherche et aussi à ses suggestions.
Je tiens à exprimer ma gratitude à nombreux membres de ma famille qui ont continué à me
soutenir et m'encourager. Je ne serai jamais en mesure de rembourser vos actes innombrables
de gentillesse et d'amour, ni de vos sacrifices personnels. Je suis particulièrement reconnaissant à
mon père et ma mère qui ont consacré plus de temps pour moi que toute autre personne.
v
RÉSUMÉ
L’analyse des coques minces soumises à un écoulement d’air supersonique a été le sujet de
plusieurs recherches. La plupart de ces études traite de l’analyse linéaire des coques cylindriques
ouvertes avec ou sans écoulement supersonique. Peu de travaux ont été faits pour l’analyse non
linéaire (géométrique et/ou du fluide) des coques cylindriques fermées soumises à l’écoulement
supersonique.
On propose dans cette thèse le développement d’un modèle qui est capable d’analyser
linéairement et non linéairement des coques minces, élastiques, anisotropes, fermées et soumises
à des pressions internes et d’écoulement d’air supersonique externe. La dynamique non linéaire et
l’instabilité de flottement des coques cylindriques circulaires fermées sont aussi analysées. La
méthode développée est une combinaison de la méthode des éléments finis, de la théorie des
coques minces et de la théorie aérodynamique du fluide. Les coques fermées, sujet de l’étude,
sont principalement simplement supportées et les autres conditions frontières arbitraires sont
aussi traitées.
La première partie de ce travail traite l’analyse non linéaire structurale (grande amplitude de
vibration) des coques cylindriques vides et fermées. La coque est divisée en plusieurs éléments
finis de type coque cylindrique fermée et les fonctions de déplacements linéaires sont dérivées de
la théorie de coque cylindrique mince de Sanders. Les coefficients modaux sont dérivés de la
théorie non linéaire de Sandres-Koiter pour les coques cylindriques minces en termes des
fonctions de déplacements linéaires développées. Les matrices de rigidité non linéaires du second
et troisième ordre sont déterminées à partir de la méthode des éléments finis. Les équations non
linéaires sont résolues par la méthode numérique de Runge-Kutta d’ordre quatre. Les fréquences
linéaires et non linéaires sont alors déterminées en fonction de l’amplitude du mouvement de la
coque pour plusieurs cas. Les résultats obtenus sont en bonne concordance avec ceux des autres
auteurs.
Dans la seconde partie de cette thèse, on présente un modèle pour l’analyse de flottement des
coques cylindriques, soumises à un écoulement supersonique externe. L’équation linéaire de la
théorie aérodynamique de Piston nous a permet d’exprimer directement la pression exercée par le
fluide comme une fonction des déplacements en tout point de la structure. L’intégration
vi
analytique de la pression nous donne des matrices exprimant l’interaction fluide-structure dans
l’écoulement supersonique. Plusieurs figures présentées dans cette partie illustrent la théorie et le
comportement dynamique des coques cylindriques, soumises à un fluide en écoulement
supersonique. Une bonne concordance des résultats a été obtenue avec d’autres théories et
expériences.
La troisième partie de cette recherche présente un modèle pour prédire l’influence des non-
linéarités associées aux grandes amplitudes de vibration de parois de la coque et à l’écoulement
supersonique du fluide des coques cylindriques minces, soumises à des pressions internes et de la
compression axiale. Avec les matrices de masse et de rigidité linéaires et non linéaires de la
coque vide ainsi qu’avec les matrices de rigidité et d’amortissements linéaires et non linéaires du
fluide en écoulement supersonique, nous développons dans cette partie des matrices linéaires et
non linéaires pour l’écoulement supersonique d’air. En dérivant une expression non linéaire pour
la pression aérodynamique en fonction des déplacements nodaux ainsi que de la combinaison des
effets non linéaires de l’écoulement supersonique du fluide et la grande amplitude de vibration de
la structure, l’équation non linéaire obtenue est résolue par la méthode numérique de Runge-
Kutta d’ordre quatre. Les fréquences linéaires et non linéaires sont alors déterminées en fonction
de l’amplitude de vibration de la coque et l’épaisseur des parois des coques pour plusieurs cas.
Cette méthode combine les avantages de la méthode des éléments finis qui traite des coques
complexes et la précision de la formulation basée sur des fonctions de déplacements dérivées de
la théorie des coques et aussi la théorie aérodynamique de Piston qui donne l’expression de la
pression aérodynamique à chaque point de la structure étudiée. On a donc ici un modèle puissant
pour prédire les caractéristiques de flottement linéaires et non linéaires des coques cylindriques
soumises à l’écoulement supersonique du fluide.
vii
ABSTRACT
Analysis of thin shells subjected to supersonic airflow has been the subject of several researches.
Most of these studies treat the linear analysis of open cylindrical shells (curved plates) with or
without supersonic flow. Few works have been done for nonlinear analysis (geometric and/or
fluid) of closed cylindrical shells subjected to supersonic flow. We propose here in this thesis to
develop a model that is able to linear and nonlinear analysis of thin elastic anisotropic closed
shells, and subjected to internal pressure and external supersonic airflow. Nonlinear dynamics
and flutter instability of closed circular cylindrical shells are also analyzed. The model developed
is a combination of finite element method, thin shell theory and aerodynamic fluid theory. Closed
shells are mainly simply supported; other arbitrary boundary conditions are also treated. The first
part of this work studies the nonlinear structural analysis (large amplitude of vibration) of empty
closed cylindrical shells. The shell is divided into several closed cylindrical finite element and the
functions of motion are derived from linear theory of thin cylindrical shells of Sanders. The
modal coefficients are derived from the nonlinear theory of Sandres-Koiter for thin cylindrical
shells in terms of linear displacement functions developed. The nonlinear stiffness matrices of
second and third order are determined from the finite element method. The nonlinear equations
are solved by fourth order Runge-Kutta numerical method. The linear and nonlinear frequencies
are then determined as a function of the movement amplitude of the shell for several cases. The
results obtained are in good agreement with those of other authors. In the second part of this
thesis, we present a model for the flutter analysis of cylindrical shells, subjected to an external
supersonic flow. The linear equation of aerodynamics Piston theory allows us to express directly
the pressure exerted by the fluid as a function of the displacements at any point of the structure.
The analytical integration of the pressure gives us matrices expressing the fluid-structure
interaction in supersonic flow. Several figures presented in this section illustrate the theory and
the dynamic behavior of cylindrical shells subjected to supersonic fluid flow. A good agreement
was obtained with other theories and experiments .The third part of this research presents a model
to predict the influence of nonlinearities associated with large amplitude of vibration of the shell
walls and the supersonic fluid flow of thin cylindrical shells taking into account the internal
pressure and axial compression. With linear and nonlinear mass and stiffness matrices of the
empty shell as well as the linear and nonlinear stiffness and damping fluid matrices, we develop
viii
consequently the linear and nonlinear matrices for supersonic airflow. By deriving an expression
for the non-linear depending on the aerodynamic pressure nodal displacements, as well as the
combination of non-linear effects of the supersonic flow of the fluid and the large amplitude of
vibration of the structure, the non-linear equation obtained is resolved by the fourth order Runge-
Kutta numerical method. The linear and nonlinear frequencies are then determined as a function
of the amplitude of vibration of and the thickness of the shells for several cases. This method
combines the advantages of the finite element method treating a complex shells and accuracy of
the formulation based on displacement functions derived from shell theory and also aerodynamic
Piston theory which gives an expression of aerodynamic pressure at each point of the structure
studied. So here we have a powerful model to predict the linear and nonlinear characteristics
during flutter phenomena of cylindrical shells subjected to supersonic fluid flow.
ix
TABLE DES MATIÈRES
DÉDICACE ................................................................................................................................... III
REMERCIEMENTS ..................................................................................................................... IV
RÉSUMÉ ........................................................................................................................................ V
ABSTRACT .................................................................................................................................VII
TABLE DES MATIÈRES ............................................................................................................ IX
LISTE DES TABLEAUX .......................................................................................................... XIII
LISTE DES FIGURES ............................................................................................................... XIV
LISTE DES SIGLES ET ABRÉVIATIONS ............................................................................. XVI
INTRODUCTION ........................................................................................................................... 1
CHAPITRE 1: REVUE DE LITTÉRATURE ................................................................................ 3
1.1 Revue de littérature .......................................................................................................... 3
1.2 Plan de la thèse ................................................................................................................. 7
CHAPITRE 2: ARTICLE 1- EFFECT OF GEOMETRIC NON-LINEARITIES ON NON-
LINEAR VIBRATIONS OF CLOSED CYLINDRICAL SHELLS ............................................... 9
2.1 Abstract ............................................................................................................................ 9
2.2 Introduction ....................................................................................................................... 9
2.3 Linear Part ...................................................................................................................... 11
2.4 Non-linear Part ............................................................................................................... 15
2.5 Numerical resolution ...................................................................................................... 19
2.6 Discussion of Results ..................................................................................................... 21
2.7 Conclusion ...................................................................................................................... 26
2.8 Appendix ........................................................................................................................ 27
x
2.9 References ...................................................................................................................... 29
CHAPITRE 3: ARTICLE 2- EFFECTS OF GEOMETRIC NONLINEAIRITIES ON
CYLINDRICAL SHELLS SUBJECTED TO A SUPERSONIC FLOW ..................................... 31
3.1 Abstract .......................................................................................................................... 31
3.2 Nomenclature ................................................................................................................. 31
3.3 Introduction .................................................................................................................... 33
3.4 Structural Model ............................................................................................................. 36
3.4.1 Linear Part .................................................................................................................... 36
3.4.2 Nonlinear Part .............................................................................................................. 39
3.5 Stress Matrix .................................................................................................................. 43
3.6 Aerodynamic Model ....................................................................................................... 44
3.7 Nonlinear Analysis ......................................................................................................... 46
3.8 Calculation and Discussion ............................................................................................ 48
3.8.1 Convergence of the Method ......................................................................................... 48
3.8.2 Onset of Flutter Predictions for Cylindrical Sheels in Supersonic Flow ..................... 49
3.8.3 Behavior of nonlinearity flutter for a pressurized cylindrical shell ............................. 50
3.8.4 Effect of Freestream Static Pressure in Post-Flutter .................................................... 51
3.8.5 Effect of internal stresses in the flutter stage of cylindrical shells subjected to
supersonic flow ...................................................................................................................... 52
3.8.6 Effect of length-radius ratios of cylindrical shells on the supersonic flutter ............... 53
3.8.7 Effect of Shell Boundary Conditions ........................................................................... 54
3.8.8 Effect of supersonic flow on the nonlinear behavior of cylindrical shells ................... 56
3.8.9 Effect of circumferential wave number ........................................................................ 56
3.9 Conclusion ...................................................................................................................... 57
3.10 Appendix ........................................................................................................................ 59
xi
3.11 References ...................................................................................................................... 62
CHAPITRE 4 : ARTICLE 3 - EFFECT OF GEOMETRIC AND FLUID FLOW
NONLINEARITIES ON CYLINDRICAL SHELLS SUBJECTED TO A SUPERSONIC FLOW
4.1 Abstract .......................................................................................................................... 64
4.2 Nomenclature ................................................................................................................. 64
4.3 Introduction .................................................................................................................... 67
4.4 Structural model development ....................................................................................... 69
4.4.1 Masse and stiffness matrices for linear structural model ............................................. 69
4.4.2 Stiffness matrices for nonlinear structural model ........................................................ 72
4.5 Initial stress matrix ......................................................................................................... 77
4.6 Aerodynamic model development ................................................................................. 78
4.7 Nonlinear numerical analysis ........................................................................................ 80
4.8 Calculation and discussion ............................................................................................. 83
4.8.1 Convergence of the method for differents modes ........................................................ 83
4.8.2 Onset of flutter predictions for cylindrical shells subjected to a supersonic flow ....... 84
4.8.3 Validation of model ...................................................................................................... 85
4.8.4 Effect of circumferential wave mode ........................................................................... 86
4.8.5 Effect of internal pressure ............................................................................................ 87
4.8.6 Effect of supersonic flow regimes in flutter of circular cylindrical shells ................... 88
4.8.7 Nonlinear flutter behavior for different longitudinal modes ........................................ 89
4.9 Conclusion ...................................................................................................................... 90
4.10 Appendix ........................................................................................................................ 92
4.11 References ...................................................................................................................... 96
xii
CHAPITRE 5: DISCUSSION GÉNERALE ................................................................................ 98
CONCLUSIONS ET RECOMMANDATIONS ........................................................................... 99
BIBLIOGRAPHIE ................................................................................................................. ….101
xiii
LISTE DES TABLEAUX
Tableau 1.1: Catégories d’analyse de flottement des coques ......................................................... 3
Tableau 3.1: Critical freestream static pressure for different shells boundary conditions,
px=0.0 psi........................................................................................................................................55
xiv
LISTE DES FIGURES
Figure 2.1: Cylindrical finite element geometry ........................................................................... 11
Figure 2.2: Finite element discretization and nodal displacement ................................................ 13
Figure 2.3: Influence of large amplitude on nonlinear frequency ................................................. 21
Figure 2.4: Influence of large amplitude on nonlinear frequency ratio for n=10 and m=1 .......... 22
Figure 2.5: Influence of longitudinal modes on nonlinear vibration at large amplitude ............... 23
Figure 2.6: Influence of various boundary conditions on nonlinear behavior of cylindrical
shells ........................................................................................................................... 24
Figure 2.7: A comparison with the results of Evensen and Ganapathi (ξ = 0.5, ε = 0.01 and 1.0);
, Present study; , Evensen [8]; , Ganapathi [15]. ............................. 24
Figure 2.8: A comparison with the results of Evensen (ξ = 2.0, ε = 0.01 and 1.0)
; Present analysis; , Evensen [8] ................................................................. 25
Figure 2.9: The amplitude-frequency relation. , Present analysis
; Ganapathi [15] ................................................................................................. 26
Figure 3.1: Cylindrical shell and finite element geometries ......................................................... 35
Figure 3.2: Finite element discretization and nodal displacement .............................................. 37
Figure 3.3: Natural frequency vs Number of elements ................................................................. 49
Figure 3.4: a) Real part and b) imaginary part of eigenvalues of the system vs freestream
static pressure p∞, n = 23, pm = 0.5 psi, px = 0.0 psi. ............................................... 50
Figure 3.5: Flutter frequency vs mean amplitude of first mode .................................................... 51
Figure 3.6: Amplitude of flutter vs frequency flutter for different regions of flutter ................... 52
Figure 3.7: Effect of internal pressure during flutter .................................................................... 53
Figure 3.8: Flutter amplitude vs frequency flutter for different L/R ............................................. 54
Figure 3.9: Flutter amplitude vs flutter frequency for different shell boundary conditions .......... 55
xv
Figure 3.10: Effect of supersonic flow in nonlinear behavior of cylindrical shells
M = 3, p∞ = 0.5 psi, pm = 0.0 psi, n = 23 .................................................................... 56
Figure 3.11: Flutter amplitude versus frequency flutter of cylindrical shells at different
circumferential wave numbers .................................................................................. 57
Figure 4.1: Finite element discretization and nodal displacement ................................................ 70
Figure 4.2: Naturel frequency vs Number of elements ................................................................. 84
Figure 4.3: a) Real part and b) Imaginary part of eigenvalues of the system vs freestream static
pressure p∞, n = 23, pm = 0.5 psi, px = 0.0 psi. ........................................................... 85
Figure 4.4: Flutter frequency vs amplitude, pec = 0.66 psi. ........................................................... 86
Figure 4.5: Amplitude flutter vs frequency flutter at n = 23 and n = 24 ....................................... 87
Figure 4.6: Effect of internal stress during the flutter, n = 25, pec =0.75 psi ................................. 88
Figure 4.7: Flutter amplitude vs frequency flutter for different L/R ............................................. 89
Figure 4.8: Flutter amplitude vs flutter frequency for different longitudinal modes .................... 90
xvi
LISTE DES SIGLES ET ABRÉVIATIONS
HOT Highly optimized tolerance
NASP National Aerospace Plante
HSCT High-Speed Civil Transport
RLV Reusable Launch Vehicle
JSF Joint Strike Fighter
QSP Quiet Supersonic Plateform
DARPA Defense Advanced Research Projects Agency
LCO Oscillation de cycle limite
MEF Méthode d’élément finis
2-D Bidimensionnel
3-D Tridimensionnel
DOF Degré de liberté
QUAD Quadrique
1
INTRODUCTION
La première observation de flottement peut être retracée au milieu des années 1940 où les
roquettes allemandes V-2 ont rencontré des problèmes de flottement qui ont abouti à secouer les
parois en métal lorsque les roquettes ont volé à une vitesse supersonique. Les défaillances
structurelles dues au flottement ont été largement prises en compte dans les avions, les navettes
spatiales, les missiles, les roquettes et les ponts à cause de la vitesse élevée d’écoulement d’air.
Le phénomène de flottement d’une coque est généralement auto-excité par des oscillations auto-
entretenues de la partie externe d'un véhicule volant lorsqu'il est exposé à un écoulement d’air le
long de sa surface. L’origine de ce genre d'instabilité structurelle est l'interaction entre
l’écoulement de fluide et le solide. L’effet de la pression aérodynamique appliquée sur la
structure conduit à la coalescence de la fréquence entre certains modes de vibration de la coque
qui représentent un symbole spécifique de flottement, ce qui résulte que la résonance de ces
modes en coalescence amplifie l'amplitude de vibration de la coque.
L’étude de flottement de la coque tombe dans le domaine de l'aéroélasticité (Bisplinghoff et al.)
dans le sens qu’elle est un résultat des interactions entre les forces d'inerties, aérodynamiques et
élastiques. En plus du flottement de la coque, il existe un autre comportement en flottement
appelé le flottement d’aile, qui s'adresse au battement des ailes en raison de la pression
aérodynamique sur les deux côtés de la structure. Le flottement de la coque se produit dans des
structures comme les panneaux de la structure d'aéronefs et des panneaux spatiaux minces qui
subissent de la pression d’air en écoulement. Les théories fondamentales et la compréhension
physique de flottement des coques peuvent être trouvées dans de nombreux livres tels que
Dowell, Librescu et quelques articles scientifiques dans la littérature.
La prévention ou la prédiction du flottement a longtemps été reconnue comme un critère essentiel
dans la conception des véhicules de vol à haute vitesse, comme High-Speed Civil Transport
(HSCT), National Aerospace Plante (NASP), X-33 Advanced Technology Demonstrator,
Reusable Launch Vehicle (RLV), Joint Strike Fighter (JSF), X-38 Spacecraft, le program Hyper-
X de X-43 et récemment le programme Quiet Supersonic Plateform (QSP) parrainé par la
DARPA. Les recherches approfondies sur des analyses et des expériences de flottement ont
fourni des séries d'outils pour les concepteurs. Aujourd'hui, grâce aux progrès impressionnants du
matériel informatique et des logiciels, les concepteurs se dirigent de plus en plus vers les outils
2
analytiques-numériques et, par conséquent, la forte demande des méthodes de calcul numérique
plus puissant, précis et efficace pour l'analyse de flottement a lancé des activités de recherche
résurgentes.
Devant le passage de transsonique à supersonique et de supersonique à hypersonique,
l’amélioration de la vitesse de vol devient un but ultime pour la conception des structures
aérospatiales. On note aussi que pour l'analyse de flottement de la coque d'un véhicule volant à
une vitesse hypersonique, la plupart des expériences et expertises accumulées par des générations
d’aéroélasticiens sur les analyses de flottement supersonique des coques cylindriques comme
l’étude approfondie présentée dans cette thèse peuvent être exploitées. Les facteurs majeurs tels
que les choix de la théorie de l'aérodynamique, linéaire/non linéaire du modèle structural
dynamique, l'approche analytique/numérique à appliquer, les considérations des effets de
paramètres du système et l’analyse des vibrations aéroélastiques hypersoniques restent les mêmes
que pour l'analyse de flottement supersonique. L’analyse linéaire du comportement dynamique
des coques cylindriques vides ou soumises à l’écoulement supersonique reste limité et ne
correspond pas aux résultats expérimentaux ce qui nous oriente vers l’étude non linéaire afin
d’expliquer et de prédire des phénomènes aéroélastiques.
Le but de cette présente étude est de développer un logiciel basé sur une formulation modale
d’éléments finis, capable de faire l’analyse linéaire et non linéaire du comportement de flottement
des coques minces sous un écoulement d’air supersonique. Une telle formulation modale est
destinée à être un outil de conception pratique pour les coques cylindriques de véhicules
supersoniques afin d’obtenir le coût (temps de calcul) réduit efficace sans perdre de précision et
assez simple pour l’application sans recours à des manipulations mathématiques sophistiquées et
aussi de prendre en compte autant que possible les considérations de la conception réelle, par
exemple l’application de pression interne et de chargement axial qui a été incorporée dans
l’algorithme. Il est également prévu que ce logiciel aidera à l’analyse de fatigue et de la
conception du contrôleur de la suppression de flottement durant le développement de véhicules
supersoniques. La revue de littérature qui suit sert un bref résumé de la recherche pertinente et
expose également la nécessité du travail effectué dans le reste de cette thèse de recherche.
3
CHAPITRE 1 : REVUE DE LITTÉRATURE ET PLAN DE LA THÈSE
1.1 Revue de littérature
Les études théoriques et expérimentales de flottement des coques ont commencé en 1950. Les
recherches et les résultats menés dans ce stade précoce ont été résumés par Fung (1960) and
Johns (1965). Après, les auteurs ont groupé une littérature théorique sur le flottement des coques
dans des catégories basiques en se basant sur les théories structurales et aérodynamiques utilisées
dans l’article de Dowell (1970), comme montré dans le Tableau 1.1. Le tableau est pris à partir de
l’article récent de Mei et al (1999). Les types d’analyse cités dans le tableau sont aussi attribués
par Gray et Mei (1993), en considérant la théorie d’analyse de flottement de la coque proposée
par McIntosh et al (1977). Une étude bibliographique sur le sujet spécifique d’analyse de
flottement de la coque a été donnée par Reed (1997) avec le support du programme NASP. Il y a
aussi des articles de revue concernant certains types d’analyse de flottement listés dans le tableau
1.1, les articles de revue concernant l’application de la méthode des éléments finis pour l’analyse
de flottement du panneau par Bismarck-Nasr (1999) et la revue de méthodes analytiques variées
incluant l’approche d’éléments finis pour les analyses non linéaires de flottement supersonique
du panneau donnée par Zhou (1999).
Tableau 1.1 : Catégories d’analyse de flottement des coques
Théorie structurale Théorie aérodynamique Nombre de mach
Linéaire Piston linéaire √2<M<5
Linéaire Écoulement potentiel
linéarisé 1<M<5
Non linéaire Piston linéaire √2<M<5
Non linéaire Écoulement potentiel
linéarisé 1<M<5
Non linéaire Piston non linéaire M>5
Non linéaire Équation d’Euler ou
Navier –Stokes
Transsonique, Supersonique
and Hypersonique
4
à cause de la simplicité mathématique de la théorie structurale linéaire, la majorité des analyses
de flottement préliminaire appartient à l’analyse linéaire de flottement de la coque. Parmi ces
travaux, on trouve : Hedgepeth (1963), Dugundji (1966) and Cunningham (1989). Comme
indiqué clairement dans l’article de Dowell (1967), l’analyse de flottement du panneau utilisant la
théorie linéaire est incapable de prendre en compte les non-linéarités structurales. La théorie
structurale linéaire indique qu’il existe une pression dynamique critique à partir de laquelle le
mouvement de la coque devient instable et se développe exponentiellement avec le temps. En
réalité, quand l’amplitude de vibration de la coque devient grande, le panneau s’étend aussi. Cette
tension dans le plan introduit des forces significatives de traction de membrane (Woinowsky-
Krieger (1959)) de sorte que la flexion de la coque est retenue dans une plage limitée, qui est
appelée oscillation de cycle limité (LCO). L’analyse linéaire de flottement de coques est utilisée
seulement pour prédire les limites de flottement. D’autres chercheurs comme Eisley (1956),
Houboult (1958), Fung (1958), and Bolotin (1963) ont utilisé la théorie structurale non linéaire
pour la détermination des limites de flottement. Plusieurs analyses non linéaires de flottement du
panneau peuvent être trouvées dans une vaste quantité de littérature, mais celles de flottement de
coques cylindriques fermées sont rares. La théorie de large déflexion de Von Karman est
largement utilisée pour tenir compte de la non-linéarité géométrique, qui a été appliquée avec
succès au problème de flottement non linéaire du panneau. En dépit de la cohérence comparative
sur laquelle la théorie structurale non linéaire a été utilisée, différentes théories aérodynamiques
linéaires ont été employées, la plus populaire adoptée est la théorie de Piston de premier ordre
quasi-stationnaire, qui est appliquée pour la vitesse d’écoulement de nombre de Mach √2. Dowell
(1967) a appliqué la théorie aérodynamique d’écoulement potentielle linéarisée pour étudier le
comportement de LCO des plaques dans un écoulement d’air avec le nombre de Mach proche de
1. La théorie aérodynamique quasi-statique d’Ackret, connue comme la théorie statique de la
bande (Cunningham), a été appliquée à l’analyse de flottement supersonique non linéaire par
Fralich (1965). Généralement, l’analyse linéaire de flottement est utilisée pour la prédiction de
limites de stabilité à travers l’analyse propre tandis que l’analyse non linéaire de flottement a
besoin d’un traitement spécial. La plupart des approches d’analyse incluent l’intégration
numérique directe, la méthode de balance harmonique et la méthode de perturbation. La première
application de l’intégration numérique directe est due à Dowell (1967) dans l’étude des
oscillations non linéaires pour des plaques simplement supportées dans le plan en flottement.
5
Ventres (1970) a recherché le comportement non linéaire de flottement des plaques encastrées par
l’intégration numérique directe. Les procédures de cette approche incluent la première
application de la méthode de Galerkin dans le domaine spatial pour les équations différentielles
partielles non linéaires du mouvement pour donner un ensemble d’équations différentielles
ordinaires couplées dépendant du temps, donc l’intégration numérique temporelle est effectuée
pour simuler la réponse de flottement. La méthode de Raleigh-Ritz, qui exige que les modes
supposés satisfont seulement les conditions limites géométriques, est une autre approche pour
l’expansion spatiale approximée pour la déflexion de la plaque. La méthode de balance
harmonique a été aussi appliquée avec succès pour l’analyse non linéaire de flottement par
certains chercheurs (1965-1987).
Cette méthode, comparée à l’intégration numérique directe, nécessite moins de temps de calcul
mais elle est extrêmement fastidieuse à mettre en œuvre. La méthode de perturbation a été
introduite pour l’analyse non linéaire de flottement par Morino (1976); un bon compromis est
trouvé entre les méthodes de perturbation et celles de la méthode de balance par Kuo (1972). En
général, pour les deux méthodes de perturbation et de balance harmonique, la déflexion du
panneau est représentée en termes de deux à six modes normaux.
Certaines limitations intrinsèques présentent des obstacles à l’application des approches
analytiques mentionnées précédemment à des analyses plus vastes. Par exemple, la méthode de
Galerkin ou la méthode de Rayleigh-Ritz exige des hypothèses raisonnables par rapport aux
fonctions de la forme de mode normal, qui doivent satisfaire les conditions limites du panneau.
Généralement, les panneaux avec des conditions de support simple, comme les limites
simplement supportées ou encastrées, sont faciles à traiter. Pour les conditions de support
complexe, même pour la combinaison des conditions de support simple, les fonctions de
déplacement situables peuvent ne pas existées ou sont trop compliquées à manipuler. Les
propriétés matérielles anisotropiques des matériaux composites ont une large application
d’ingénierie, donc il est difficile pour analyser le flottement non linéaire utilisant les méthodes
analytiques. Compte tenu de ces problèmes, les chercheurs ont eu recours à des méthodes
numériques comme la méthode des éléments finis (MEF) dont son utilisation pour l’analyse
linéaire de flottement du panneau a été initialisée par Olson (1967), suivie par plusieurs
chercheurs (1970-1977). L’application de la MEF pour le flottement non linéaire du panneau a
commencé en 1977 par Mei dans l’étude de flottement de panneau 2D avec la négligence de
6
l’inertie de membrane. La MEF pour l’analyse non linéaire de flottement peut être en plus
classifiée en deux catégories :
La formulation d’éléments finis dans le domaine fréquentiel et la formulation dans le domaine
temporel. La formulation du domaine fréquentiel est bien développée et largement appliquée pour
résoudre les limites de flottement (analyse propre) et l’étude des oscillations harmoniques LCO.
Mei et Rogers (1976) ont incorporé le module de l’analyse de flottement supersonique pour le
panneau 2D dans le code de Nastran. Rao and Rao (1980) ont recherché le flottement
supersonique des panneaux 2D restreints dans les bornes. Mei et Weidman (1977) ont été les
premiers qui ont étudié le LCO du panneau 3D, les effets d’amortissement, les rapports d’aspect,
les forces initiales dans le plan et les conditions limites ont été examinées aussi. Le flottement des
panneaux 3D a été étudié par Mei et Wang (1982) et Hang and Yang (1983) ont utilisé les
éléments finis de plaque triangulaire pour étudier des harmoniques LCO. En comparaison, la
formulation du domaine du temps qui implique l’intégration numérique est moins documentée
dans l’analyse non linéaire de flottement des panneaux. Les principaux obstacles pour
l’implantation de la formulation dans le domaine temporel sont (Green and Killey (1972) et
Robinson (1990)) : i) le grand nombre de degrés de liberté (DOF) de système, ii) les matrices de
rigidité non linéaires pour chaque élément doit être à chaque pas de temps et iii) le pas de temps
de l’intégration est extrêmement petit.
D’autres groupes de chercheurs ont focalisé leurs études sur le flottement non linéaire du panneau
considérant la non-linéarité aérodynamique donnée par Chandiramani et Librescu (1996). Le
flottement dans l’écoulement supersonique des panneaux en composites cisaillés et déformés a
été recherché. La théorie de déformation de cisaillement et les charges aérodynamiques basées
sur la théorie de Piston de troisième ordre ont été utilisées. Les équations de flottement du
panneau ont été dérivées en utilisant la méthode de Galerkin. La méthode de continuation de la
longueur de l’arc a été utilisée pour déterminer l’état d’équilibre statique et ses stabilités
dynamiques ont été ensuite examinées. Les effets des petites imperfections géométriques du
panneau, la direction de l’écoulement d’air et la compression uniforme dans le plan sur les limites
de flottement ont été recherchés. Il a été conclu que la théorie de déformation de cisaillement et la
théorie aérodynamique non linéaire sont nécessaires pour la détermination des limites de
flottement des panneaux en composite modérément épaisses. Le mouvement de post-flottement
de ces panneaux en composite a aussi été étudié par l’application d’une technique de prédicateur-
7
correcteur de Newton-Raphson pour la solution périodique et l’intégration numérique pour les
solutions quasi-périodiques de flottement.
1.2 Plan de la thèse
La thèse inclut cinq chapitres qui répondent à la structuration d’une thèse présentée sous forme
d’articles, où chaque article contient résumé, introduction, revue de la littérature, méthode suivie,
résultats, discussion et liste des références. Après ces articles, on donne une discussion générale
et une conclusion en illustrant la réussite des objectifs visés dans ce projet de thèse. Les articles
suivants ont été soumis pour publication au cours de l’étude doctorale.
R. Ramzi, A. Lakis. Effect of Geometric Non-linearities on Non-linear Vibrations of Closed
Cylindrical Shells, accepté pour publication dans: International Journal of Structural Stability and
Dynamics.
R. Ramzi, A. Lakis. Effects of Geometric Nonlinearities on Cylindrical Shells Subjected to a
Supersonic Flow, accepté pour publication dans: Computers & Fluids.
R. Ramzi, A. Lakis. Effect of Geometric and Fluid Flow Nonlinearities on Cylindrical Shells
Subjected to a Supersonic Flow, accepté pour publication dans: International Journal of
Numerical Methods and Applications.
Le premier chapitre présente une revue de littérature pertinente dans le sujet et un plan de thèse
proposé dans ce mémoire.
Dans le chapitre 2, qui correspond au 1er
article, une formulation d’éléments finis détaillée est
donnée, ainsi que l’équation du mouvement dérivé basée sur le principe de travail virtuel. La
théorie des coques minces de Sanders-Koiter est utilisée pour aborder la non-linéarité structurale
et la méthode des éléments finis pour discrétiser la coque cylindrique. Une transformation modale
est effectuée sur l’équation de mouvement du système assemblé. Les procédures d’évaluation des
matrices de rigidité non linéaires sont présentées aussi.
Dans le chapitre 3, qui correspond au 2e article, les équations gouvernant le mouvement en
coordonnées modales sont transformées en une représentation d’état d’espace afin de faciliter la
mise en œuvre de l’intégration numérique directe. La méthode d’intégration de Runge-Kutta
d’ordre quatre est aussi appliquée pour trouver des résultats de l’analyse des coques cylindriques
soumises à l’écoulement supersonique.
8
Le chapitre 4, qui correspond au 3e article, présente des résultats numériques et des discussions
d’une analyse non linéaire géométrique des coques cylindriques et aussi de l’écoulement du
fluide. Les amplitudes de vibration sont comparées aux résultats existants des méthodes
analytiques et numériques à des fins de vérification.
Une discussion générale vient de montrer la validation de la méthodologie suivie et les résultats
obtenus par la comparaison avec différentes études des recherches publiées par d’autres
chercheurs pour chaque article considéré dans cette thèse et ceci est le sujet du 5e chapitre.
À la fin de cette thèse, une conclusion générale et des recommandations ont été données pour
récapituler tous les sujets abordés dans cette thèse et le travail futur prévu.
9
CHAPITRE 2: ARTICLE 1 - EFFECT OF GEOMETRIC NON-
LINEARITIES ON NON-LINEAR VIBRATIONS OF CLOSED
CYLINDRICAL SHELLS
Submitted, 22 march 2012 to the International Journal of Structural Stability and Dynamics
Redouane Ramzi and Aouni Lakis
2.1 Abstract
Geometrically non-linear free vibrations of closed isotropic cylindrical shells are investigated
through an analytical-numerical model. The method developed is a combination of Sanders-
Koiter nonlinear shell theory and the finite element method. The cylindrical shell is subdivided
into cylindrical finite elements and the displacement functions are derived from exact solutions of
Sander’s equations for thin cylindrical shells. Expressions for the mass, linear and non-linear
stiffness matrices are determined by exact analytical integration. Various boundary conditions of
shell and in-plane effects are considered. Non-linear responses are analyzed using the Runge-
Kutta numerical method. The non-linear frequency ratio is determined with respect to the
amplitude thickness ratio of the motion for different study cases. Detailed numerical results are
presented for various parameters for a closed isotropic shell. The results indicate either hardening
or softening types of non-linear behaviors, depending on the structure data.
The results of this approach show good agreement with pertinent results published by many
authors considering several cases of shells. This research clarifies the current disagreement about
types of cylindrical shells with geometric non-linearities.
2.2 Introduction
In recent development work for the aerospace, spatial, oil and nuclear industries, the study of
thin-walled cylindrical shells is of great importance in structural design to obtain the desired
performance, rigidity, lightness, strength and stability characteristics of a structure in different
conditions. This challenge has led to several published studies based on thin shell theories such
us Sanders-Koiter, Donnell’s shallow-shell, Flũgge-Lur’e-Byrne and Novozhilov theories. The
resonant frequency in non-linear vibration is a function of its amplitude through the frequency-
amplitude relationship which indicates whether the non-linearity is hardening or softening. The
10
first study on non-linear vibrations of circular cylindrical shells was undertaken by Reissner[1] in
1955. Donnell’s shallow-shell theory for a single half-wave of the vibration mode was applied to
the case of simply supported shells, and the results showed that the non-linearity could be either
of the hardening or softening type. Cummings [2] has remarked that the governing equations
used by Reissner depend on the area of integration. Chu [3] made a similar analysis but for a
closed cylindrical shell, and found that the non-linearity characteristic was of the hardening type.
Evensen [4] experimentally tested some shells that were subjected to large vibration amplitudes
and observed that the non-linearity was of the softening type. After this he re-examined Chu [3]
and Nowinski’s [5] analysis. Olson [6] also observed a slight softening non-linearity during
experimental studies of a thin seamless shell made of copper. Evensen and Fulton [7] studied the
non-linear dynamic response of thin circular cylindrical shells, and they concluded that the non-
linearity may be either hardening or softening depending upon the ratio of the number of axial
waves to the number of circumferential waves. Evensen [8] analyzed the free and forced non-
linear vibrations of thin circular rings using two vibration modes (driven and companion modes),
and applying Galerkin’s procedure for resolution of the equations of motion. Evenseen [9] also
studied infinitely long cylindrical shells using the harmonic balance method. The non-linear free
vibration of circular cylindrical shells was examined by Alturi [10] using Donnell’s equations
given by Chu [3], and Dowell and Ventres [11]. The Galerkin technique was used to reduce the
problem to a system of coupled non-linear ordinary differential equations for the modal
amplitudes. The result was a non-linear hardening type. Chen and Babcock [12] analyzed the
large vibration amplitudes of a simply-supported thin-walled cylindrical shell using the
perturbation method to study the influence of the companion mode on the non-linear forced
response of a circular cylindrical shell. The results indicate that the non-linearity may be either
softening or hardening depending on the mode. To account for geometric non-linearity, small
arbitrary initial imperfections, and using the strain-displacement relations of the Sanders-Koiter
non-linear theory, Radwan and Genin [13-14] derived a set of non-linear modal equations for thin
shells of arbitrary geometry using the modal expansion method. The results obtained indicated a
hardening type non-linearity. Using a QUAD-4 shear flexible developed shell element model
(FEM), the free non-linear vibrations of thin circular cylindrical shells have been studied by
Ganapathi and Vardan [15]. The non-linear governing equations were solved using a Welson
numerical integration scheme; modified Newton-Raphson iterations have been employed.
11
The approach presented in this article involves two steps:
Using linear strain-displacement and stress-strain relationships which are inserted into Sander’s
equations of equilibrium, we determine the displacement functions by solving the linear equation
system. We then determine the mass and linear stiffness matrices for a finite element and
assemble the matrices for the complete shell.
Using strain-displacement relationships from the Sanders-Koiter non-linear theory, the modal
coefficients are obtained from the displacement functions. The second and third order non-linear
stiffness matrices for a finite element are then calculated by precise analytical integration with
respect to modal coefficients.
The finite element proposed was a closed cylindrical segment (Figure 1) with four degrees of
freedom at each node: axial, radial, circumferential, and rotational.
Figure 2.1: Cylindrical finite element geometry
This approach is more accurate than usual finite element methods. Other advantages offered by
this method include:
Arbitrary boundary conditions can be used - High and low frequencies may be obtained with high
accuracy - Precise results are obtained in cases with low and high circumferential wave numbers -
Valid for a large amplitude of vibration that can reach 5 times the wall thickness of the closed
cylindrical shell - More accurate than the usual finite element methods - Computation time is
very short -Structured and simple method.
2.3 Linear Part
The equations of motion of an anisotropic cylindrical shell in terms of U, V, and W (axial,
tangential and radial displacements) (Figure 2) are written as follows:
0),,,( ijJ PVWUL (1)
W
U
V
12
Where ( 1,2,3)J JL
are three linear partial differential equations (given in Appendix) and ijp are
elements of the elasticity matrix for anisotropic shells which is given by:
11 12 14 15
21 22 24 25
33 36
41 42 44 45
51 52 54 55
63 66
0 0
0 0
0 0 0 0[ ]
0 0
0 0
0 0 0 0
p p p p
p p p p
p pP
p p p p
p p p p
p p
For the isotropic case,
0 0 0 0
0 0 0 0
(1 )0 0 0 0 0
2[ ]
0 0 0 0
0 0 0 0
(1 )0 0 0 0 0
2
D D
D D
D
PK K
K K
K
where,
3
212(1 )
EhK
Bending stiffness,
21
EhD
Membrane stiffness
The displacement functions associated with the circumferential wave number are assumed in the
normal manner, as:
( , ) cos
( , ) cos
( , ) sin
x
R
x
R
x
R
U x Ae n
W x Be n
V x Ce n
(2)
(3)
(4)
13
Figure 2.2: Finite element discretization and nodal displacement
where n is the circumferential mode,
Note that this approach treats closed cylindrical shells based on the circumferential wave number,
not on the longitudinal wave number.
Substituting (4) into the equations of motion (1), a system of three homogeneous equations, linear
functions of constants A, B, C, are obtained. For the solution to be non-trivial, the determinant of
this system must be equal to zero. This brings us to the following characteristic equation:
8 6 4 2
8 6 4 2 0Det H f f f f f
The values of coefficients if in this eighth-degree polynomial are given in Appendix.
Each root of this equation yields a solution to the linear equations of motion (1). The complete
solution is obtained by adding the eight solutions independently with constants Ai, Bi and Ci
(i = 1, 2,…., 8). These constants are not independent. We can therefore express Ai and Bi as a
function of Ci, as follows:
i i i
i i i
A C
B C
The values of i and i can be obtained from system (1) by introducing relation (6). To
determine eight iC constants, it is necessary to formulate eight boundary conditions for the finite
elements. The axial, tangential and radial displacements will be specified for each node. The
following relation defines the element displacement vector at the boundaries:
(5)
(6)
W
U
W V
un
dwn/dx wn
xj=l
xi=0
14
Tjjjiiij
iwvuwvu
i
j
A C
where the elements of matrix [A] are given in Appendix.
Multiplying Eq. (8) by [A]-1
and substituting the result into Eq. (4), we obtain:
j
iN
,xW
,xV
,xU
where [N] represents the shape function matrix given in this form:
1N T R A
and [T], [R], [A] are described in Appendix.
The constitutive relation between the stress and deformation vector of a cylindrical shell is given
as:
j
iBP
]][[
The mass and stiffness matrices for each element are derived using the classic finite element
procedure and can then be expressed as:
dA]B][P[]B[]k[
dA]N[]N[h]m[
TL
T
Where is the density of the shell, h its thickness and dA Rdxd . The matrices [m] and [k] are
obtained analytically by carrying out the necessary matrix operations over x and θ in the
equation. The global matrices sM and sK may be obtained, respectively, by assembling the
mass and stiffness matrices for each individual finite element.
(11)
(12)
(10)
(7)
(8)
(9)
15
2.4 Non-linear part
The coefficients of the modal equations are obtained using the Lagrange method in the same way
developed by Radwan and Genin [13], paying attention to geometric non-linearities. Following
this, the second and third non-linear stiffness matrices corresponding to geometric non-linearities
are computed and superimposed on the linear system.
Shell displacements (u,v,w) are expressed as sums of generalized product coordinates iq t and
spatial functions U, V, W (see Eq.(4)).
,
,
,
i i
i
i i
i
i i
i
u q t U x
v q t V x
w q t W x
The deformation vector is written as a function of the generalized coordinates by separating the
linear part from the non-linear part.
, ,2 , , ,2T
L NL xx x xx x
Where subscripts ``L`` and ``NL`` mean ``linear`` and ``non-linear``, respectively, and L and
NL are defined in Appendix.
The Lagrange’s equations of motion in generalized coordinates iq t are found using Hamilton’s
principle and Eq. (13):
iiii
V
q
T
q
T
dt
d
where T is the total kinetic energy, V is the total elastic strain energy of deformation and the iQ ’s
are the generalized forces. In this case study; 0iQ (free vibration).
Using the vector deformation given by (14), the developed formulation for total kinetic and strain
energy and inserting these into the Lagrange equation (15), the following non-linear modal
equations are obtained after a long development of equations:
(13)
(14)
(15)
16
( ) ( 2) ( 3) 0
1,2,....
L NL NL
pr r pr r prs r s prsq r s q
r r r s r s q
m k k k
p
Where prm , L
prk are terms of the mass and linear stiffness matrices given by Eq. (12). The terms
( 2)NL
prsk and ( 3)NL
prsqk represent the second and third-order non-linear stiffness and are given by the
following integrals for the case of an isotropic closed cylindrical shell:
dACp)ED(pBpApK prsprsprsprsprs)NL(
prs 331222112
and
( 3)
11 22 12 33( )NL
prsq prsq prsq prsq prsq prsqK p A p B p D E p C dA
Where dA Rdxd , pij are the terms of the elasticity matrix [P], and the terms
Aprs,Bprs,Cprs,Dprs,Eprs and Aprsq,Bprsq,Cprsq,Dprsq,Eprsq represent the coefficients of the modal
equations. These coefficients are given by the following equations:
Aprs=ap Ars+ar Asp+as Apr Aprsq=2Apq Ars
Bprs=bp Brs+br Asp+bs Bpr Bprsq=2Bpq Brs
Cprs=cp Crs+cr Csp+cs Cpr Cprsq=2Cpq Crs
Dprs=ar Bsp+as Bpr+bp Ars Dprsq=2Apq Brs
Eprs=br Asp+bs Apr+as Brs Eprsq=2Bpq Ars
with;
ap = Up,x , bp= 1/R ( Vp,θ + Wp ), cp= 1/2 ( Up,θ / R + Vp,x )
Apq =1/8 R2 ( R Vp,x – Up,θ ) . ( R Vq,x – Uq,θ ) + 1/2 Wp,xWq,x
Bpq = 1/8 R2 ( R Vp,x – Up,θ ) . ( R Vq,x –Uq,θ ) + 1/2 R2 ( Wp,θ - Vp ) . ( Wq,θ - Vq )
Cpq = 1/4 R ( Wp,x Wq,θ – Wq,x Wp,θ ) – 1/4 R ( Vp Wq,x + Vq Wp,x )
where U, V and W are spatial functions determined by Eq. (4).
In Eqs. (16 - 20), the subscripts ‘p,q’, ’p,r,s’ and ‘p,r,s,q’ represent the coupling between two,
three and four modes, respectively.
(16)
(17)
(18)
(19)
(20)
17
Let us illustrate the development of the expression for the (p,q) term of matrices Apq and Apqrs.
The expression Apq is given by:
/p q x R
pq pqA a e
and Apqrs is :
8 8( ) /
1 1
p r s q x R
prsq rs pq rs
r s
A a a e
rs is the term (r,s) of matrix [E], where [E] represents a matrix of constants defined by
1T
E A A , and
1 22 2
2
1 1sin cos
2 4pq pq pqa a n a n
R
with
1
pq p p p q q qa n n
2
pq p qa
Similarly, the matrices [Bpq]… [Eprsq] can be written as a function of α, β, λ, x and θ. The (p, q)
terms of these matrices are described in Appendix. For details of the basis of the presented
method, consult Ref. [13].
After integration, we obtain:
2 21 1TNL NL
prsk A J A
3 31 1
28
TNL NL
prsqk A J AR
(21)
(22)
(23)
(24)
(25)
(26)
(27)
18
The (p,q) term in matrix 2
,
NL
p qJ
and 3
,
NL
p qJ
are written as:
8 8
1 1
2
8 8
1 1
1 , 1
0,
1 ,
0
p q rsr R
l s p q r
NL p q r
sr
k l
p q r
G p q e
ifJ p q
G p qR
if
and
8 8
1 1
3
8 8
1 1
2 , 1
0,
2 ,
0
p q r ssr R
k l p q r s
NL p q r s
sr
k l
p q r s
G p q e
ifJ p q
G p qR
if
where G1(p,q) and G2(p,q) are coefficients in conjunction with α, β, λ and elements pij in matrix
[P]. The general expressions of G1(p,q) and G2(p,q) are given by :
1 1 1 1 1 11 1 1
(p,q) 11 1
1 2 1 2 1 21 1 1
11 2
1 2 1 2 1 21 1 1
22 1
1 1 1 1 1 11 1 1
33 1
1 11
12 1
G1 = p rs r sp s pr
p rs r sp s pr
p rs r sp s pr
p rs r sp s pr
r sp
p II a A a a A a a A a
p II a A a a A a a A a
p II b A b b A b b A b
p II c A c c A c c A c
p II a A b
1 1 1 11 1
1 21
12 2
1 1 1 1 1 11 1 1
12 1
1 2 1 21 1
12 2
s pr p rs
p rs
r sp s pr p rs
r sp s pr
a A a b A a
p II b A a
p II b A a b A a a A b
p II b A a b A a
where,
3
1
sin 2
3
nII
2
2
sin(2 )(2 cos (2 ))
3
n nII
n
(28)
(29)
(30)
(31)
19
and
1 1 2 2 1 1
( , ) 11 22 12 11 22
2 1 1 2 2 1 1 2
12 11 12
1 1 1 1
12 22
1 1 2 2 2 1 1 2
33
32 2 3
16
1
4
3
4
1
4
p q pl kq pl kq pl kq
pl kq pl kq pl kq pl kq
pl kq pl kq
pl kq pl kq pl kq pl kq
G p p p a a p a a p b b
p a b b a p p a a a a
p p a b b a
p c c c c c c c c
where the terms 1a and 2
a are given by Eqs. (24-25). Terms b(1)
…, c(1)
…and c(2)
are
coefficients appearing in expressions for the elements of matrices [Bpq] and [Cpq] defined in Eqs
(20). These coefficients are given in Appendix.
2.5 Numerical resolution
The mass and stiffness matrices, both linear and non-linear, given in Eqs. (12-26-27) are only
determined for one element. After subdividing the shell into several elements, the global mass
and stiffness matrices are obtained by assembling the matrices for each element.
Assembly is done in such a way that all the equations of motion and the continuity of
displacements at each node are satisfied. These matrices are designated as [M], [K(L)
], [K(NL2)]
and [K(NL3)
] respectively.
2 32 3 0L NL NL
M K K K
Where δ is the displacement vector and M , L
K
, 2NL
K
and 3NL
K
are respectively
the mass, linear and second and third-order non-linear stiffness matrices of the system.
In practice, very specific conditions are applied to the shell boundaries. Thus, these square
matrices are reduced to square matrices of order NREDUC=NDF*(N+1)-J, where J represents
the number of constraints applied. The system of equations then becomes:
2 3
2 3 0r r r r r r r r
L NL NLM K K K
Where the subscript ‘‘r’’ means, ‘‘reduced’’.
(32)
(33)
(34)
20
Let us set:
rq
where represents the square matrix for eigenvectors of the linear system and q is a time-
related vector. Substituting Eq. (33) into system (32) and pre-multiplying the whole equation by
T
, we obtain;
2 3
2 3 0T T T Tr r r r
L NL NLM q K q K q K q
The products of matrix T r
M
and T r
LK
represent diagonal matrices, written
as D
M
and D
LK
respectively. By neglecting the cross-product terms in 2
q and
3
q of Eq. (34), we obtain:
2 32 2
1 1
0NREDUC NREDUC
L NL NL
pp p pp p ps s ps s
s s
m q k q k q k q
Thus there are ‘‘NREDUC’’ simultaneous equations in the form of Eq. (35).
Setting p p pq t A f with 0 1pf and 0 0pf
Eq. (35) becomes, after the iA simplification and dividing byppm ,
2 3
23 2 3/ / 0
L NL NL
pp pp pp
pp p p p p p
pp pp pp
k k kf f t A t f t A t f
m m m
where t represents the shell thickness. The coefficient /pp ppk m represents the p linear vibration
frequency of system. The solution pf of these ordinary non-linear differential equations
which satisfies the initial conditions (36) is approximated by a fourth-order Runge-Kutta
numerical method. The linear and non-linear natural angular frequencies are evaluated using a
systematic search for the pf roots as a function of time. The /NL L ratio is expressed as a
function of the non-dimensional ratio /pA t where pA is the vibration amplitude.
(39)
(38)
(35)
(36)
(37)
21
2.6 Discussion of Results
The single mode response of an empty, closed isotropic cylindrical shell corresponding to m=1
and n=4, is shown in Figure 2.3. The calculation has been performed for a simply supported
cylindrical shell with the following dimensions and material properties: t=0.0254 cm, R=2.54 cm,
L=40 cm, E=200 GPa, ν=0.3, ρ=7800 kg/m3. It illustrates a non-linearity of the hardening type
for a low number of circumferential waves. The presented results are compared with Refs. [22-
23]. Ref. [22] was obtained based on Donnell’s simplified non-linear method considering only
lateral displacement. Raju and Rao used the finite element method based on an energy
formulation.
It is observed that the variation ratio between the non-linear and linear frequency Wnl/Wl
increases as the ratio A/t increases although these variations are small for A/t below 1.0. The
difference between results obtained by the present theory and those of Refs. [22-23] might be due
to the fact that Nowinski [22] neglected in-plane inertia and took into account only lateral
displacement. Raju and Roa [23] expressed the displacement components along the shell
generator in polynomial form.
Figure 2.3: Influence of large amplitude on non-linear frequency
In Figure 2.4, the non-linear frequency response as a function of the amplitude of vibration for
the non-linear free vibration case for Olsen’s shell (n=10, m=1, ξ=0.1635, ε=0.003025, ν=0.365)
is compared with theoretical and experimental results obtained previously by several authors in
order to validate the present work. This case concerned an empty isotropic cylindrical shell with
0,9
0,95
1
1,05
1,1
1,15
1,2
1,25
1,3
1,35
1,4
0 1 2 3 4
Nowinski
Rajuand Rao
my study
Wnl/W
l
A/t
22
simply-supported ends. The geometrical and material characteristics of the cylindrical shell
considered by Olson [6] are: L=0.39 m, R= 0.203 m, t = 0.11176x10-3
m, E = 124x109 Pa,
ν =0.35 and ρ = 8930 kg/m3. Good agreement can be seen, which allows us to confirm that the
new approach presented in this paper can be used in analyzing the influence of large amplitude
vibrations on simply-supported closed cylindrical shells.
Figure 2.4: Influence of large amplitude on non-linear frequency ratio for n=10 and m=1
In Figure 2.5, the effect of the axial longitudinal mode on non-linear behavior of the cylindrical
shell under large-amplitude vibration for circumferential wave number n = 10, is investigated.
Here, the closed shell is simply supported and has the following data: E = 200 GPa, v=0.3, ρ =
7800 kg/m3, R = 2.54 cm, L = 40 cm, t = 0.0254 cm. Results show that the non-linearity is the
softening type for m=1 and the hardening type for other longitudinal axial modes. A more
pronounced non-linearity is seen at mode m = 7.
0
0,5
1
1,5
2
2,5
3
0,99 0,995 1 1,005 1,01
Rougi[21]
Evensen[8]
chen/babcock[12]
amabil[24]
my study
experimental study[6]
A/t
Wnl/Wl
23
Figure 2.5: Influence of longitudinal modes on non-linear vibration at large amplitude
An advantage of the presented model is that it allows the study of various boundary conditions.
This is clearly shown in Figure 2.6, which presents the effect of boundary conditions on non-
linear free vibration. The shell tested has the same data as Figure 2.5.
The non-linearity is the hardening type for the shell with free-free boundary conditions, however
the softening type is observed for other boundary conditions, and more pronounced in clamped-
clamped and simply supported – simply supported. The effect is small for a cylindrical shell
under clamped-simply supported and clamped-free boundary conditions.
0,8
0,85
0,9
0,95
1
1,05
1,1
1,15
1,2
1,25
0 1 2 3 4
m=1m=2m=3m=4m=5m=6
Wnl/W
l
A/t
0,8
0,85
0,9
0,95
1
1,05
1,1
1,15
1,2
1,25
1,3
0 0,5 1 1,5 2 2,5 3 3,5
m=7
m=8
m=9
m=10
m=11
m=12
Wnl/W
l
A/t
24
Figure 2.6: Influence of various boundary conditions
on nonlinear behavior of cylindrical shells
As a next step; a detailed parametric study was carried out for the simply-supported shell, and
results from the presented method were compared with those of available analytical models. Two
values of aspect ratio ξ = mπR/nL and three values of parameter ε = (n2h/R)
2 were considered.
They are; ξ = 0.5 and ε = 0.01, 1.0; m and n are the axial half wave and circumferential wave
numbers and L and R are the length and radius of the cylindrical shell respectively.
Figure 2.7: A comparison with the results of Evensen and Ganapathi
(ξ = 0.5, ε = 0.01 and 1.0);
, Present study; , Evensen [8]; , Ganapathi [15].
0
0,5
1
1,5
2
2,5
3
3,5
0,999 0,9995 1 1,0005 1,001
C-C
C-F
C-S
S-F
S-S
F-F
A/t
Wnl/Wl
0
0,4
0,8
1,2
1,6
0,75 0,8 0,85 0,9 0,95 1
Є=1.0 Є=0.01
A/t
Wnl/Wl
25
In Figure 2.7, the results of our study show that the non-linearity is the softening type for small
values of ξ (for example ξ = 0.5), and as ε increases the vibrations become more and more non-
linear. It can be concluded that the strength of the non-linearity depends on the value of the
parameter ε. For comparison purposes, Evensen’s and Ganapathi’s results [8-15] have been
plotted in Figure 2.7. The result obtained using the present method represents a good
compromise. Furthermore, in Figure 2.8 a hardening behavior of non-linearity vibrations is
observed for large values of ξ (ξ = 2.0 as shown in Figure 2.8). Similar observations are revealed
using the Evensen analysis [8]; the non-linearity trend becomes stronger for large values of ε.
The numerical results obtained in this section for ξ = 0.5 and ξ = 2.0 are also in good agreement
with results reported in Ref. [25].
Figure 2.8: A comparison with the results of Evensen (ξ = 2.0, ε = 0.01 and 1.0).
; Present analysis; , Evensen [8]
To study the influence of in-plane boundary conditions on non-linear vibrations of a cylindrical
shell, the case of a simply-supported shell with axial displacement is restrained (u = w = v = 0).
Figure 2.9 presents the results obtained in comparison with Ganapathi analysis 15, for various
values of ξ and ε. Analysis of Figures [2.7-9] reveals that the trends of non-linearity behavior
does not change, which is confirmed by Ganapathi’s [15] results for an isotropic cylindrical shell.
0
1
2
3
4
5
6
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6
Є=0.1 Є=0.01
A/t
Wnl/Wl
26
Figure 2.9: The amplitude-frequency relation. , Present analysis
; Ganapathi [15]
2.7 Conclusion
The non-linear free vibrations of thin-walled cylindrical shells were analyzed using an approach
based on non-linear Sanders-Koiter shell theory and a hybrid finite element method. The second
and third order stiffness matrices for an empty closed cylindrical element were considered. This
study was performed for various shells examined in the literature for both low and high
circumferential wave numbers. A precise prediction of non-linearity trends under high-amplitude
vibration can be provided using the presented method. The effects of various boundary conditions
and axial modes on non-linear vibration behaviors have also been treated. In all cases, the degree
of non-linearity is dependent upon thickness, radius, length, circumferential and axial wave
numbers. However, the type of non-linearity is determined by the aspect ratio ξ. The results
obtained for non-linearities are in good agreement with available studies for ξ = 0.5, 2.0. and ε =
0.01, 0.1, 1.0. In-plane boundary conditions have no effect the nature of non-linearity. Results
for low and high wave numbers compare closely with previous studies based on different non-
linear shell theories and numerical methods for non-linear equation resolution. The presented
method allows a large field study of geometric and material parameters under various conditions
in which these structures may be used.
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
0,85 0,9 0,95 1 1,05 1,1 1,15 1,2 1,25
ξ=2
Є=1.0
ξ=2
Є=0.01
ξ=0.5
Є=1.0
ξ=0.5
Є=0.01
A/t
Wnl/Wl
27
2.8 Appendix
Sander’s linear equations for thin cylindrical shells in terms of axial, tangential and
circumferential displacements are:
The coefficients of characteristic Eq. (5) are given by:
2
1 33 36 66
2
3 12 33 15 36 66
5 15 36 66
2
7 22 55 25
2
9 33 36 66
11 36 24 66 54
4
3 4
2
2
3 9 4
2 3 /
f p p R p R
f p p p p R p R
f p p p R R
f p p R p R
f p p R p R
f p p p R p R R
2928 11 44 14
22 2 2 2
26 9 1 44 11 45 11 66 5 14 7 11 44 14 11 11 3 44 3 11 14
9 11 24 14 12
42
24 1 7 44 9 11 55 45 66 1 9 7 11 3 25 55 3 14 11 11
2 4 2 2
2
12 4 . 2 2
ff p p p
r
nf f f p p p p p f rp f p p p r f p f p rf f pr
f p p p pr
nf f f p f p p p p f f f p f p p f p f p rrr
2 2
11 3 5 1 11 5 7 14 5 9 25 22 3 14 11 11
2 2
12 5 9 7 14 3 11 24 3 1 9 7 11 9 11 25 9 11 22 12
6 2 2 2 2
2 7 45 66 55 1 9 7 11 3 5 7
25 55 1 11
(2 ) (2 ) ( ). 2
2 2 2
2 4
1 . 2 2
f r f f f f rf f p rf f n r p rp f r p f p
p f f r f p f f r p f f f f p f p p f p p p
f n r f p p p f f f p f r f f
p r p rf f rf
3 5 11 25 11 55
4 2
1 7 24 25 1 9 7 11 3 12 5 7 3 25 3 55
2
25 22 1 11 11 25 11 55 3 5
2 2 2
22 1 9 7 11 3 25 22 11 25 11 22 3 12 7 12
4
0 1 7 22
1
2 2 2
2 1 1
1 1 2
2
f p p r p p
n r f f p p f f f p f p rf f f p f r p
p rp f f r p p r p p f f
n p f f f p f r p rp r p p p p f p f p
f n f f p r n
2
2 4 2 2 3
25 55 1 25 55 25 221p n r p n f n r p r p n r p rp
2 2 2 3 3 2 2 2
15 33 63121 11 142 2 3 2 2 2 2
3 2 2
36 66
2 3 2 2
2
5121 22 2
1( , , , ) ( ) ( ) ( )( )
2
3 1( )( 2 )
2 2 2
1( , , , ) ( ) (
ij
ij
P P PPU U V W W W V V UL U V W P P P
x R x x x x R x x R R x R
P P W V U
R R x x R
PP PUL U V W P
R R x R
2 3 3 2
52 54 25 552 24
2 2 2 2 2 2 3 2
2 2 3 2 2
63 6633 362 2 2
3 3 2
423 41 43 2 2
1)( ) ( ) ( )( )
3 31 3 1( )( ) ( )( 2 )
2 2 2 2
( , , , ) ( )ij
P P P PPV W W W V
R R R R x R R R
P PV U W V UP P
R x R x R R x x R x
PU V WL U V W P P P
x R x x
4 4 3 3 3
45 634 4 2 2 2 2 2 2
4 3 3 3 3 2 4 3
66 51 52 55
2 2 2 2 2 2 2 3 3 2 4 4 3
4
5421 22
2 2 2 2
2 1( ) ( )
2 3 1( 2 ) ( ) ( )
2 2
( )
P PW W V U V
x R x x R R x x
P P P PW V U U V W W V
R x x R x R x R R
PP PU W VW
R x R x R
2 2
2524
2 2 3 2( )
PP W W V
R R
28
Matrices 3 3[ ]T , 3 8[ ]R , and 8 8[ ]A are defined as:
/ / /
/
/ / /
cos( ) 0 0
[ ] 0 cos( ) 0
0 0 sin( )
(1, ) , (2, ) , (3, ) 1,2,...,8
(1, ) , (2, ) 1, (3, ) , (4, ) , (5, )
(6, ) , (7, ) , (8, )
i i i
i
i i i
x R x R x R
i i
l Rii i i
l R l R l Rii
n
T n
n
R i e R i e R i e i
A i A i A i A i A i eR
A i e A i e A i e iR
1,2,...8
Expressions L and NL for cylindrical shells are given by:
2
2
2
2 2
2
2
1
1
1
2 3 1
2 2
L
U
x
VW
R
V U
x R
W
x
W V
R
W V U
R x R x R
and
2 2
2 2
2
1 1 1
2 8
1 1 1
2 8
1
2
0
0
0
NL
W V U
x x R
W V UV
R x R
W W WV
R x x
Matrices pqB and
pqC
/p q x R
pq pqB b e
/p q x R
pq pqC c e
where
1 1 2
2
1 1sin
2 4pq pq pqb a b n
R
1 2
2
1sin cos
4pq pq pqc c c n n
R
with
1
pq p p p q q qa n n
1
pq p qb n n
1
pq p qc n , 1,...,8p q
29
2.9 References
1. E. Reissner, Non-linear Effects in Vibrations of Cylindrical Shells, Ramo-Wooldridge
Corporate Report. (1995) AM5-6.
2. B.E. Cummings, Some non-linear vibration and response problems of cylindrical panels and
shells, Graduate aeronautical Laboratories, California Institute of technology, Pasadona. (1962)
SM 62-32 (AFOSR 3123).
3. H.N. Chu, Influence of large amplitudes on flexural vibrations of thin circular cylindrical
shells, Journal of aerospace Sciences. 28 (1961) 602-609.
4. D.A. Evensen, Some observations on the non-linear vibration of thin cylindrical shells,
American Institute of Aeronautics and Astronautics Journal. 1(1966) 2857-2858.
5. J. Nowinski, Non-linear transverse vibrations of orthotropic cylindrical shells, American
Institute of Aeronautics and Astronautics. 1 (1963) 617-620.
6. M.D. Olson, Some experimental observations on the non-linear vibration of cylindrical shell,
American Institute of Aeronautics and Astronautics 3 (1965) 1775-1777.
7. D.A. Evensen and R.E. Fulton, Some studies on the non-linear dynamic response of shell-type
structures, Proceedings of an International Conference on Dynamic Stability of Structures,
Evanston, IL 237-254, Oxford & New York: Pergamon Press (1966).
8. D.A. Evensen, Nonlinear flexural vibrations of thin-walled circular cylinders, Nasa Technical
Note, Langley Research Center, (1967) NASA- TN D-4090 .
9. D.A. Evensen, Non-linear vibrations of an infinitely long cylindrical shell, American Institute
of Aeronautics and Astronautics Journal. 6 (1968) 1401-1403.
10. S. Atluri, A perturbation analysis of non-linear free flexural vibrations of a circular
cylindrical shell, International Journal of Solids and Structures. 8(1972) 549-569.
11. E.H. Dowell and C.S. Ventres, Modal equations for the non-linear flexural vibrations of a
cylindrical shell, International Journal of Solids and Structures. 4 (1968) 975-991.
12. J.C. Chen and C.D. Babcock, Non-linear vibration of cylindrical shells, American Institute of
Aeronautics and Astronautics Journal, 13 (1975) 868-876.
30
13. H. Radwan and J. Genin, Non-linear modal equations for thin elastic shells, International
Journal of Non-linear Mechanics. 10 (1975) 15-29.
14. H.R. Radwan and J. Genin, Non-linear vibrations of thin cylinders. Journal of Applied
Mechanics, 43(1976) 370-372.
15. M. Ganapathi and T.K. Varadan, Large amplitude vibrations of circular cylindrical shells,
Journal of sound and vibration. 192 (1996), 1-14.
16. H.V Ramani, An analytical investigation of large amplitude free flexural vibration of thin
circular cylindrical shells, M.S. Dissertation (1985), Indian Institute of Technology, Madras,
India.
17. A.A. Lakis and M.P. Paidoussis, Dynamic Analysis of Axially Non-Uniform Thin Cylindrical
Shells, Journal of Mechanical Engineering Science. 14(1) (1972) 49-71.
18. A. Selmane, and A.A. Lakis, Non-Linear Dynamic Analysis of orthotropic Open Cylindrical
Shells Subjected to a Flowing Fluid, Journal of Sound and Vibration. 202(1) (1997) 67-93.
19. J.L. Sanders, An Improved First-Approximation Theory for Thin Shell. (1959) NASA R-24.
20. M.H. Toorani, and A.A. Lakis, General equations of Anisotropic Plates and Shells Including
Transverse shear Deformations, Rotary Inertia and Initial Curvature Effects, Journal of Sound
and Vibration. 237(4) (2000) 561-651.
21. R. Rougui, and F. Moussaoui, R. Benamar, Geometrically non- linear free and forced
vibrations of simply supported circular cylindrical shells: A semi-analytical approach,
International Journal of Non-linear Mechanics. 42 (2007) 1102-1115.
22. J.L. Nowinski, Non-linear transverse vibrations of orthotropic cylindrical shells, AIAA . 1
(1963) 617-620.
23. K.K, Raju, G.V. Rao, Large amplitude asymmetric vibrations of some thin elastic shells, Int.
J. Non-linear Mech. 10 (1975) 15-29.
24. M. Amabili, F. Pellicano, M.P. Paidoussis, Non-linear vibrations of simply supported circular
cylindrical shells coupled to quiescent fluid, J. Fluids Struct. 12 (1998) 883-918.
31
CHAPITE 3: ARTICLE 2 - EFFECTS OF GEOMETRIC
NONLINEAIRITIES ON CYLINDRICAL SHELLS SUBJECTED TO A
SUPERSONIC FLOW
Submitted, 15 June 2012 to Computers and Fluids
Redouane Ramzi and Aouni Lakis
3.1 Abstract
A nonlinear analysis is presented for cylindrical shells subjected to supersonic flow using
Sanders-Koiter nonlinear shell theory for large amplitude of vibrations and linear piston theory
for modeling the fluid/structure interaction. Internal pressure and axial loads are taken into
account. A hybrid finite element is developed and applied for discretizing the system. Linear and
nonlinear mass, stiffness and damping matrices are found. The governing equation of the system
is resolved using the Runge Kutta numerical method. The flutter amplitude of vibration versus
the flutter frequency is obtained. Several parameters that are critical in the design of aeronautic
structures are investigated here. Results show that the system loses stability due to coupled mode
flutter; however a softening type of nonlinearity in all cases studied occurs beyond the flutter.
The results are in good agreement with existing theoretical and experimental results for flutter
frequency and amplitude published in the literature.
3.2 Nomenclature
a∞ Freestream speed of sound
Ai Motion amplitude
[A] Matrix defined in Appendix
[B] Defined by Equation 11
[Cf] Aerodynamic damping matrix of fluid
ap,bp,cp Modal coefficients determined by equation 20
apq,bpq,cpq Coefficients determined by equations in Appendix
a(i)
pq,b(i)
pq,c(i)
pq Coefficients determined by equations in Appendix
Apq, Bpq, Cpq Modal coefficients determined by equations in Appendix
Aprs, Bprs, Cprs , Dprs, Eprs Modal coefficients determined by equation 19
Aprsq, Bprsq, Cprsq, Dprsq, Eprsq Modal coefficients determined by equation (19, 22)
32
D Membrane stiffness, defined by Equation 3
{ε} Deformation vector giving in Appendix
E Young’s modulus
[E] Matrix defined by Equation 23
{Fp} Force vector due to the aerodynamic pressure field
h Shell thickness
K Bending stiffness of shell, defined by Equation 3
[Ks] Global stiffness matrix for a shell
[k] Stiffness matrix for a shell element
[kf] Local aerodynamic stiffness matrix
[kI] Initial stiffness matrix for a shell element
[Kf] Global aerodynamic damping matrix
[KI] Global initial stiffness matrix for a shell
[KNL2
] Nonlinear stiffness matrix of second order
[KNL3
] Nonlinear stiffness matrix of third order
L Shell length
M Mach number
[Ms] Global mass matrix for a shell
n Circumferential number
[m] Mass matrix for a shell element
N Cylindrical shell number
Nx,Nθ,Nxθ Stress resultant for a circular cylindrical shell
xN ,N Stress resultant due to shell internal pressure and axial compression
[N] Defined by Equation 10
pa Aerodynamic pressure
p∞ Freestream static pressure
pm, px Shell internal pressure and axial compression
[P] Elasticity matrix
Qx, Qθ Transverse stress resultant for a circular cylindrical shell
R Shell radius
[Rf] Matrix defined in Appendix
33
U Axial displacement
Ui Potential energy due to initial strain
V Circumferential displacement
W Radial displacement
x Longitudinal coordinate
δi, δj Displacement at node i and j
λj Complex roots of the characteristic equation
υ Poisson’s ratio
γ Adiabatic exponent
θ Circumferential coordinate
ρ Shell density
ρf Fluid density
ω Oscillation frequency
øxx, øxθ, øn Elastic rotations for a circular cylindrical shell
{σ} Stress vector
3.3 Introduction
The aeroelastic stability of closed cylindrical shells in supersonic flow is of keen interest for
designers of skin panels of missiles, the space shuttle, and military aerospace vehicles. The first
event of flutter instability on these structures appears to have been on the V-2 rocket. Among
authors who focus their studies on developing linear models to investigate shell stability in
supersonic flow, we find: Dowell, Olson and Fung, Barr and Stearman, and Ganapathi et al.
However, a major difference occurs between experimental results and those from linear studies
for prediction of flutter and the behavior of structures during flutter. A nonlinear study therefore
becomes necessary to explain the behavior observed in the experiments. Very few studies have
addressed the nonlinear aeroelastic stability of cylindrical shells subjected to supersonic flow.
In 1955, Librescu [1,2] was the first to study the stability of circular cylindrical finite shallow
shells. He used Donnell’s nonlinear shallow shell theory and piston theory for calculating the
supersonic flow pressure, taking into account only simple mode expansion. No results on limit
cycle amplitudes were given. A simplified form of Donnell’s nonlinear shallow shell theory is
applied by Olson and Fung [3] for simply-supported shells considering a simple two-mode
34
expansion including an axisymmetric term. The supersonic flow/structure interaction was
modeled using linear piston theory. Evensen and Olson [5] considered the companion mode and
four-degree-of-freedom mode expansion, which allows the study of travelling-wave flutter. The
same behavior of travelling waves is predicted and measured for large-amplitude forced vibration
of shells. Following this study, linear theoretical and experimental studies of supersonic flutter of
circular cylindrical shells including geometric imperfections were published by Carter and
Stearman [6] and Barr and Stearman [7,8]. They also developed a linear piston theory to calculate
the supersonic flow pressure for Mach numbers superior to 1.6. Vol’mir and Medvedeva [9],
studied the nonlinear supersonic flutter of circular cylindrical shells, considering the initial
deflection and axial loads. They used a combination of Donnell’s nonlinear shallow shell theory
and linear piston theory; the finite difference numerical method was applied to search the
solutions.
Amabili and Pellicano [10] investigated the aeroelastic stability of simply-supported, circular
cylindrical shells in supersonic flow using Donnell’s nonlinear shallow-shell theory describing
large amplitude and linear piston theory to modulate the fluid flow/structure interaction. Their
study takes into account structural damping and employs the Galerkin Method for descretizing
the system. Results show that the system loses stability by standing wave flutter through
supercritical bifurcation, however a very small increment in the free streamstatic pressure can
produce a travelling wave flutter.
Horn et al [11] conducted an experimental study of the stability of circular cylindrical shells in
axial air-flow of both subsonic and supersonic regimes. Divergence was found for subsonic flow
and flutter for supersonic flow. Another experimental study of supersonic flutter of pressurized
and axially compressed circular cylindrical shells was performed at the NASA Ames Research
Center, California, in 1962 and 1964 [12].
The approach presented in this paper is summarized in three points:
- Using linear strain-displacement and stress-strain relationships which are inserted into
Sander’s equations of equilibrium, we determine the displacement functions by solving the
linear equation system. We then determine the mass and linear stiffness matrices for a finite
element and assemble the matrices for the complete shell.
35
- Using strain-displacement relationships from Sanders-Koiter nonlinear theory, the modal
coefficients are obtained from the displacement functions. The second and third-order
nonlinear stiffness matrices for a finite element are then calculated by precise analytical
integration with respect to modal coefficients.
- Linear piston theory is applied for modeling and determining the supersonic flow pressure
and describing the structure/fluid interaction.
The equations of cylindrical shells are used to obtain the displacement functions instead of using
the more common arbitrary polynomial forms. The finite element proposed is a closed cylindrical
segment (Figure 3.1) with four degrees of freedom at each node: axial, radial, circumferential,
and rotational.
i = 1, 2… N
Figure 3.1: Cylindrical shell and finite element geometries
The approach developed here is more accurate than usual finite element methods. Different
arbitrary boundary conditions can be used, and high and low frequencies may be obtained with
high accuracy. Precise results are obtained with low and high circumferential wave numbers.
Large amplitude vibration can be examined. The presented approach also enables short
computation time and higher precision of results due to use of exactitude piston theory, which
yields the pressure created by supersonic air flow at each point in the structure.
x 1
2 3
N
U
W
V
i
36
3.4 Structural Model
3.4.1 Linear Part
The equations of motion of an anisotropic cylindrical shell in terms of U, V and W (axial,
tangential and radial displacements) (Figure 3.2) are written as follows:
0),,,( ijJ PVWUL
Where ( 1,2,3)JL J are three linear partial differential equations (given in Appendix) and ijp are
elements of the elasticity matrix for anisotropic shells which is given by:
11 12 14 15
21 22 24 25
33 36
41 42 44 45
51 52 54 55
63 66
0 0
0 0
0 0 0 0[ ]
0 0
0 0
0 0 0 0
p p p p
p p p p
p pP
p p p p
p p p p
p p
For the isotropic case,
0 0 0 0
0 0 0 0
(1 )0 0 0 0 0
2[ ]
0 0 0 0
0 0 0 0
(1 )0 0 0 0 0
2
D D
D D
D
PK K
K K
K
where,
3
212(1 )
EhK
Bending stiffness,
21
EhD
Membrane stiffness
The displacement functions associated with the circumferential wave number are assumed in the
normal manner, as:
( , ) cos
( , ) cos
( , ) sin
x
R
x
R
x
R
U x Ae n
W x Be n
V x Ce n
(2)
(1)
(3)
(4)
37
Figure 3.2: Finite element discretization and nodal displacement
where n is the circumferential mode,
Note that this approach treats closed cylindrical shells based on the circumferential wave number,
not on the longitudinal wave number.
Substituting (4) into the equations of motion (1), a system of three homogeneous equations, linear
functions of constants A, B, C, is obtained. For the solution to be non-trivial, the determinant of
this system must be equal to zero. This brings us to the following characteristic equation:
8 6 4 2
8 6 4 2 0Det H f f f f f
The values of coefficients if in this eighth-degree polynomial are given in Appendix.
Each root of this equation yields a solution to the linear equations of motion (1). The complete
solution is obtained by adding the eight solutions independently with constants Ai, Bi and Ci (i =
1,2,….,8). These constants are not independent. We can therefore express Ai and Bi as a function
of Ci, as follows:
i i i
i i i
A C
B C
W
U
W
V
un
dwn/dx
wn
xj=l
xi=0
(5)
(6)
38
The values of i and i can be obtained from system (1) by introducing relation (6). To
determine eight iC constants, it is necessary to formulate eight boundary conditions for the finite
elements. The axial, tangential and radial displacements will be specified for each node. The
following relation defines the element displacement vector at the boundaries:
{ } { }
CAj
i
where the elements of matrix [A] are given in Appendix.
Multiplying Equation (8) by [A]-1
and substituting the result into Equation (4), we obtain
j
iN
,xW
,xV
,xU
where [N] represents the shape function matrix given in this form:
and [T], [R], [A] are described in Appendix.
The constitutive relation between the stress and deformation vector of a cylindrical shell is given
as:
j
iBP
]][[
The mass and stiffness matrices for each element are derived using the classic finite element
procedure and can then be expressed as:
dA]B][P[]B[]k[
dA]N[]N[h]m[
T
T
Where is the density of the shell, h its thickness and dA Rdxd . The matrices [m] and [k] are
obtained analytically by carrying out the necessary matrix operations over x and θ in the
(11)
(12)
1N T R A (10)
(7)
(8)
(9)
39
equation. The global matrices sM and sK may be obtained, respectively, by assembling the
mass and stiffness matrices for each individual finite element.
3.4.2. Nonlinear Part
The coefficients of the modal equations are obtained using the Lagrange method in the same way
developed by Radwan and Genin [13], paying attention to geometric nonlinearities. Following
this, the second and third nonlinear stiffness matrices corresponding to geometric nonlinearities
are computed and superimposed on the linear system.
Shell displacements (u, v, w) are expressed as sums of generalized product coordinates iq t and
spatial functions U, V, W (see Eq. (4)).
,
,
,
i i
i
i i
i
i i
i
u q t U x
v q t V x
w q t W x
The deformation vector is written as a function of the generalized coordinates by separating the
linear part from the nonlinear part.
, ,2 , , ,2T
L NL xx x xx x
Where subscripts ``L`` and ``NL`` mean ``linear`` and ``nonlinear``, respectively, and L and
NL are defined in Appendix.
The Lagrange’s equations of motion in generalized coordinates iq t are found using Hamilton’s
principle and Equation (13):
iiii
V
q
T
q
T
dt
d
where T is the total kinetic energy, V is the total elastic strain energy of deformation and the iQ ’s
are the generalized forces.
Using the vector deformation given by (14), the developed formulation for total kinetic and strain
energy and inserting these into the Lagrange equation (15), the following nonlinear modal
equations are obtained after a long development of equations:
(13)
(14)
(15)
40
( ) ( 2) ( 3) 0
1,2,....
L NL NL
pr r pr r prs r s prsq r s q
r r r s r s q
m k k k
p
Where prm , L
prk are terms of the mass and linear stiffness matrices given by Equation (12).
The terms )NL(
prsk 2 and
)NL(prsqk
3 represent the second and third-order nonlinear stiffness and are
given by the following integrals for the case of an isotropic closed cylindrical shell:
dACp)ED(pBpApK prsprsprsprsprs)NL(
prs 331222112
and
( 3)
11 22 12 33( )NL
prsq prsq prsq prsq prsq prsqK p A p B p D E p C dA
Where dA Rdxd , ijp are the terms of the elasticity matrix [P], and the terms Aprs, Bprs, Cprs,
Dprs, Eprs and Aprsq, Bprsq, Cprsq, Dprsq, Eprsq represent the coefficients of the modal equations.
These coefficients are given by the following equations:
Aprs=ap Ars+ar Asp+as Apr Aprsq=2Apq Ars
Bprs=bp Brs+br Asp+bs Bpr Bprsq=2Bpq Brs
Cprs=cp Crs+cr Csp+cs Cpr Cprsq=2Cpq Crs
Dprs=ar Bsp+as Bpr+bp Ars Dprsq=2Apq Brs
Eprs=br Asp+bs Apr+as Brs Eprsq=2Bpq Ars
with
ap = Up,x , bp= 1/R ( Vp,θ + Wp ), cp= 1/2 ( Up,θ / R + Vp,x )
Apq =1/8 R2
( R Vp,x – Up,θ ) . ( R Vq,x – Uq,θ ) + 1/2 Wp,xWq,x
Bpq = 1/8 R2 ( R Vp,x – Up,θ ) . ( R Vq,x –Uq,θ ) + 1/2 R
2 ( Wp,θ - Vp ) . ( Wq,θ - Vq )
Cpq = 1/4 R ( Wp,x Wq,θ – Wq,x Wp,θ ) – 1/4 R ( Vp Wq,x + Vq Wp,x )
Where U, V and W are spatial functions determined by Equation (4).
(16)
(17)
(18)
(19)
(20)
41
In Equations (16) - (20), the subscripts ‘p,q’, ’p,r,s’ and ‘p,r,s,q’ represent the coupling
between two, three and four modes, respectively.
Let us illustrate the development of the expression for the (p,q) term of matrices Apq and Apqrs.
The expression Apq is given by:
/p q x R
pq pqA a e
and Apqrs is :
8 8( ) /
1 1
p r s q x R
prsq rs pq rs
r s
A a a e
rs is the term (r,s) of matrix [E], where [E] represents a matrix of constants defined by
1T
E A A , and
1 22 2
2
1 1sin cos
2 4pq pq pqa a n a n
R
with
1
pq p p p q q qa n n
2
pq p qa
Similarly, the matrices [Bpq]… [Eprsq] can be written as a function of α, β, λ, x and θ. The (p, q)
terms of these matrices are described in Appendix. For details of the basis of the presented
method, consult reference [13].
After integration, we obtain:
2 21 1TNL NL
prsk A J A
3 31 1
28
TNL NL
prsqk A J AR
The (p,q) terms in matrix 2
,
NL
p qJ
and 3
,
NL
p qJ
are written as:
(21)
(22)
(23)
(24)
(25)
(26)
(28)
(27)
42
8 8
1 1
2
8 8
1 1
1 , 1
0,
1 ,
0
p q rsr R
l s p q r
NL p q r
sr
k l
p q r
G p q e
ifJ p q
G p qR
if
and
8 8
1 1
3
8 8
1 1
2 , 1
0,
2 ,
0
p q r ssr R
k l p q r s
NL p q r s
sr
k l
p q r s
G p q e
ifJ p q
G p qR
if
where G1(p,q) and G2(p,q) are coefficients in conjunction with α, β, λ and elements pij in matrix
[P]. The general expressions of G1(p,q) and G2(p,q) are given by :
1 1 1 1 1 11 1 1
11 1
1 2 1 2 1 21 1 1
11 2
1 2 1 2 1 21 1 1
22 1
1 1 1 1 1 11 1 1
33 1
1 11
12 1
G1(p,q) = p rs r sp s pr
p rs r sp s pr
p rs r sp s pr
p rs r sp s pr
r sp
p II a A a a A a a A a
p II a A a a A a a A a
p II b A b b A b b A b
p II c A c c A c c A c
p II a A b
1 1 1 11 1
1 21
12 2
1 1 1 1 1 11 1 1
12 1
1 2 1 21 1
12 2
s pr p rs
p rs
r sp s pr p rs
r sp s pr
a A a b A a
p II b A a
p II b A a b A a a A b
p II b A a b A a
where,
3
1
sin 2
3
nII
2
2
sin(2 )(2 cos (2 ))
3
n nII
n
(29)
(30)
(31)
(32)
(33)
43
and
1 1 2 2 1 1
11 22 12 11 22
2 1 1 2 2 1 1 2
12 11 12
1 1 1 1
12 22
1 1 2 2 2 1 1 2
33
32( , ) 2 3
16
1
4
3
4
1
4
pl kq pl kq pl kq
pl kq pl kq pl kq pl kq
pl kq pl kq
pl kq pl kq pl kq pl kq
G p q p p p a a p a a p b b
p a b b a p p a a a a
p p a b b a
p c c c c c c c c
where the terms 1
a and
2a are given by Equations (25-26). Terms b
(1)…, c
(1)…and c
(2) are
coefficients appearing in expressions for the elements of matrices [Bpq] and [Cpq] defined in
Equations (20). These coefficients are given in Appendix.
3.5 Stress Matrix
The potential energy is given as:
dxRdNNNNUl
nxxxxi
0
2
0
222 )(2/1
where;
RpN
R
pN
m
xx
2
xN , N are stress resultants due to shell internal pressure mp and axial compression xp
respectively, l is the element length, is about the normal to the x plane, xx is the strain
rotation about x , and n is the rotation about the normal to a shell element.
This rotation vector is expressed as [14],
)1
(2
1
)(1
x
VU
R
WV
R
x
W
n
xx
(34)
(35)
(37)
(36)
44
Upon insertion of displacements (U,W,V) given in Equation (8), the potential energy is rewritten
in terms of the nodal degrees of freedom, which leads to the initial stiffness matrix for each
element in the following form:
dxRdNC
NN
N
N
CNkl
TT
x
x
TT
I
0
2
000 ][][
00
00
00
][][][
where;
xR
R
x
C
2
10
2
1
01
0
00
][ 0
3.6 Aerodynamic Model
The pressure on the body surface is approximated by the pressure exerted by a piston moving
in the flow with a velocity equivalent to the superposition of the velocity due to the changing of
body shape and the velocity due to the rigid-body motion of the body:
12
211
/
a/
p
p
where;
p is the aerodynamic pressure,
p∞ is the freestream static pressure,
γ is the adiabatic exponent,
a∞ is the freestream speed of sound
ω is the velocity of the piston.
The piston velocity ω is given by [18,19]:
x
WU
t
W
U is the freestream velocity and W is the radial displacement.
(40)
(41)
(39)
(38)
45
Considering the linear terms in the binomial expansion of Equation (40), we obtain:
t
W
ax
WMPPa
1
The form of the radial displacement assumed is the same as that considered by Lakis and
Paidoussis in [15]:
8
1
8
1 j
j
j
)tR
x(i
W)ncos(eWj
Where j is the jth
root of the characteristic equation, is the natural angular frequency and n is
the circumferential wave number.
Substituting Equation (43) into Equation (42), we obtain the equation for the aerodynamic
pressure on the cylindrical shell wall:
j
if
j
if
j
radiala ARTa
pARTMp
Ripp
11
0
0
The damping and stiffness for fluid flow are then obtained by substituting Equations (9) and (44)
into the integral (45);
[ ] { }T
p aF N p dA
The linear matrix of aerodynamic damping [Cf] and stiffness matrix [Kf] are given as follows:
1 1T
f fC A D A
1 1T
f fK A G A
where;
dxRRrPa
Dl
fT
f
0
dxRRrMPr
iG f
lTj
f
0
(45)
(43)
(44)
(42)
(46)
(48)
(49)
(47)
46
3.7 Nonlinear Analysis
The motion of a cylindrical shell subjected to an external supersonic flow is governed by the
equation of motion which may be presented as follows:
2 3
3( 2) 0i i i i iL L L NLNL
s f
j j jj j
M C K K K
where subscripts s and f refer to the shell in vacuo and in fluid flow respectively, and is the
degrees of freedom vector for the total nodes.
Further useful expressions are:
L L
s I fK K K K
dxRdNC
NN
N
N
CNK TT
x
xl
TTI
0
0
2
0
0
00
00
00
T
sK B P B dA
1 1T
L
f fK A G A
where;
T Tk
f f
iG p M R T T R dA
R
Let us set:
q
where represents the square matrix for eigenvectors of the linear system and q is a time-
related vector. Substituting Equation (55) into system (50) and pre-multiplying the whole
equation by T
, we obtain;
2 32 3
0D D D T TL L L NL NL
s fM q C q K q K q K q
(50)
(51)
(52)
(54)
(55)
(56)
(53)
47
where;
D TL L
s sM M ,
D TL L
f fC C ,
D TL L
K K
D denotes diagonal, and the matrices quantifying the fluid contribution to the matrix equations of
motion are non-symmetric. To facilitate the analysis, therefore, we consider only the symmetric
portion of the matrices. This simplifying hypothesis is valid since the original and simplified
systems have comparable dynamic behavior [20].
By neglecting the cross-product in 2
q of Equation (56), we obtain;
2 32 3
1
0NREDUC
L L NL NL
ii i ii i ii i ij j ij j
j
m q c q k q K q K q
Here we have ‘‘NREDUC’’ simultaneous equations of the form of Equation (58).
At first, we limit ourselves to solving Equation (56) by taking into account only the diagonal
terms of the products 22T NL
K
and 33T NL
K
, therefore Equation (58) is written
in this form:
2 32 3 0L L NL NL
ii i ii i ii i ii i ii im q c q k q K q K q
Setting; i i iq A f
with 10 if and 00 if
Equation (59) becomes, after the iA simplification and dividing by iim :
2
2 2 3
20i i
ii i i i i i i i i
A Af f f f f
t t
where;
ii
L
iii
m
c ;
ii
L
iii
m
k2 ;
tm
K
ii
NL
iii
2
;
3
2
NL
ii
i
ii
Kt
m
and t represents the shell thickness. The square root of coefficient /ii iik m represents the ith
linear
vibration frequency of system. The solution if of these ordinary nonlinear differential
(61)
(62)
(59)
(60)
(58)
(57)
48
equations which satisfies the initial conditions (60) is approximated by a fourth-order Runge-
Kutta numerical method. The linear and nonlinear natural angular frequencies are evaluated using
a systematic search for the if roots as a function of time. The /NL L ratio is expressed as a
function of the non-dimensional ratio /iA t , where iA is the vibration amplitude.
3.8 Calculation and Discussion
The influence of nonlinearities associated with the wall of the shell on the vibration
characteristics of closed cylindrical shells subjected to linear supersonic flow is expressed by
Equation (59).
For a shell with given characteristics, we first present the results for convergence of the model,
and the ratio Wnl/Wl of linear and nonlinear frequencies is graphically represented in Figures 3.5
to 3.10 with respect to the non-dimensional ration Ai/t.
The shell and flow properties are given by:
6 10 2
2 4 3
16 10 / (11 10 / )
0.35
0.0040 (0.0001015 )
15.4 (0.381 )
8.00 (0.203 )
0.000833 / (8900 / )
3.00
8400 / (213 / )
s
E lb in N m
h in m
L in m
R in m
lb s in kg m
M
a in s m s
The influence of nonlinearities associated with large amplitude of cylindrical shells subjected to a
fluid at supersonic flow is the main focus of this analysis.
3.8.1 Convergence of the Method
A first calculation was made to determine the required number of finite elements for precise
determination of the results, which is very important in finite element analysis. Calculations were
made for a closed circular cylindrical shell subjected to supersonic flow for the number of finite
elements N = 2, 4, 6, 8...25. The shell is simply supported at both ends. The results for n = 23 and
for m = 1, 2 and 3, are shown in Figure 3.3. Convergence of natural frequency is reached for 20
elements.
49
Figure 3.3: Natural frequency vs Number of elements
3.8.2 Onset of Flutter Predictions for Cylindrical Shells in Supersonic Flow
Figure 3.4, shows the eigenvalues associated with the first and second longitudinal modes (m=1,
2) with n = 23 (circumferential wave number) vs freestream static pressure p∞. The real part of
the eigenvalues gives the oscillation frequency, and the imaginary part gives the damping for
positive values and instability for negative values. Flutter begins where the imaginary part
crosses the zero and becomes negative. It can be seen that the shell loses stability at pcr = 0.75 psi,
a)
Figure 3.4: a) Real part and b) imaginary part of eigenvalues of the system
vs freestream static pressure p∞, n = 23, pm = 0.5 psi, px = 0.0 psi.
220
240
260
280
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
1st mode
2nd mode
3rd mode
Number of elements
Fre
quen
cy (
Hz)
548
549
550
551
552
553
554
555
556
557
558
0 0,2 0,4 0,6 0,8 1 1,2
Freestream Static Pressure (psi)
Rea
l ei
gen
val
ue
50
b)
Figure 3.4 (suite): a) Real part and b) imaginary part of eigenvalues of the system
vs freestream static pressure p∞, n = 23, pm = 0.5 psi, px = 0.0 psi.
which is called the critical freestream static pressure. This instability is known as coupled-mode
flutter. This results in good agreement to Amabili results [10,21], which is found pcr = 0.77 psi,
and coincide with value obtained by Evensen and Olson [4,5].
3.8.3 Behavior of nonlinearity flutter for a pressurized cylindrical shell
Figure 3.5 shows how the flutter frequency varies with amplitude for the first longitudinal mode
(m=1). The flutter frequency is seen to decrease slowly with increasing amplitude (softening type
of nonlinearity), which is in very good agreement with the study by Olson and Fung [3]. M=3,
pm=0.5 psi, p∞=0.66 psi; n=23; knowing that the nonlinearity is considered by large amplitude
vibrations of cylindrical shell subjected to linear supersonic flowing fluid.
-1
0
1
2
3
4
5
6
7
8
0 0,2 0,4 0,6 0,8 1
Freestream Static Pressure (psi)
Imag
inar
y e
igen
val
ue
(Hz)
51
Figure 3.5: Flutter frequency vs mean amplitude of first mode
3.8.4 Effect of freestream static pressure during the flutter
The shell tested is simply supported at both ends, and pressurized; the pressure exerted by the
supersonic flow is considered a multiple of critical freestream static pressure to study the
nonlinear behavior of flutter of a cylindrical shell during the period following the onset of the
flutter. Figure 3.6 shows how the flutter frequencies vary with flutter amplitude. The flutter
frequency is seen to decrease slowly with increasing amplitude. This is typical of the so-called
‘‘softening’’ type of nonlinearity, and the tendency is more pronounced at high supersonic flow
pressure as seen in p∞ = 1.5 pcr.
0
0,2
0,4
0,6
0,8
1
1,2
0,96 0,97 0,98 0,99 1 1,01
Olson and Fung
My study
Nondimensional flutter frequency, Wnl/Wl
Mae
n a
mp
litu
de
of
firs
t m
od
e, A
52
Figure 3.6: Amplitude of flutter vs frequency flutter for different regions of flutter
3.8.5 Effect of internal stresses in the flutter stage of cylindrical shells
subjected to supersonic flow
The effect of internal pressure on the nonlinear flutter stage is shown in Figure 7. Two values of
supersonic flow pressure p∞ = 1.5 pcr and p∞ = 3.0 pcr, that exceed the critical freestreem static
pressure (given the flutter pcr = 0.75 psi) are investigated here. It can be observed that the effect
of internal pressure becomes stronger at supersonic flow pressures (p∞ = 2.25 psi) that are larger
than the critical freestream static pressure. Otherwise, when the pm = 0.5 psi (pressurized case),
the flutter frequency gradually decreases with increasing vibration amplitude, which explains the
stabilizing effect of internal pressure applied to cylindrical shells in supersonic flow. This
confirms the conclusion of Amabili and Olson and Fung in their researches. It is also noted that
in the region near the onset of flutter (p∞ = 1.5 pcr), no difference was observed according to the
Figure 3.7. Also the softening type nonlinearity of the shell is still observed.
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
0,5 0,7 0,9 1,1
0.9pcr
pcr
1.5pcr
Wnl/Wl
A/t
53
Figure 3.7: Effect of internal pressure during flutter
3.8.6 Effect of length-radius ratios of cylindrical shells on the supersonic
flutter
Figure 3.8 shows the variation of flutter amplitude versus flutter frequency of cylindrical shells
under supersonic flow in the region corresponding to onset of flutter and during flutter stages,
depending on each length-radius case. On one side, the trend of the nonlinearity is a softening
type for L/R = 4 and 2, and a hardening type for the L/R = 1. The same behavior is observed for
supersonic flow pressure ( p∞ = 2 psi ). On the other side, the critical freestream static pressures
increase when L/R ratios decrease, and the trend of flutter frequency variation vs flutter
amplitude is more pronounced for L/R = 2. For case L/R = 1, the nonlinearity is a hardening type
and may indicate that the nonlinear geometry of the cylindrical shell modeled creates a critical
area, in which the flutter frequency is superior to natural frequency.
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10
pm = 0.0 psi , p∞ = 1.5 pcr
pm = 0.5 psi , p∞ = 1.5 pcr
pm = 0.0 psi , p∞ = 3 pcr
pm = 0.5 psi , p∞ = 3 pcr
Wnl/Wl
A/t
54
Figure 3.8: Flutter amplitude vs frequency flutter for different L/R
3.8.7 Effect of Shell Boundary Conditions
According to Table 3.1, simply-supported - free shells experience early onset of flutter at low
values of supersonic flow pressure (pcr = 0.37 psi). This can be compared to the case of shells
clamped at both ends; flutter instability occurs at higher values of supersonic flow
pressure (pcr = 1.07 psi).This clarifies the effect of boundary conditions, which can delay the
onset of flutter.
In order to illustrate the effect of boundary conditions on cylindrical shell flutter, the mode (n =
23, m = 1) for the same pressurized closed shell with different boundary conditions is shown in
Figure 3.9. The shell with simply-supported – free boundary conditions is the one with the more
pronounced nonlinearity effect. This effect is small for a panel that is simply supported at both
ends.
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
2,2
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6
L/R=4,pcr=0,18 psi
L/R=2,pcr=0.54 psi
L/R=1,pcr=1.5 psi
L/R=4,p∞=2 psi
L/R=2,p∞=2 psi
L/R=1,p∞=2 psi
Wnl/Wl
A/t
55
Table 3.1: Critical freestream static pressure for different shells boundary
conditions, px=0.0 psi
pcr (psi) ncritical pm (psi)
Clamped at both ends 0.84 23 0
Clamped at both ends 1.07 23 0.5
Free simply supported at both ends 0.75 23 0.5
Free simply supported at one end, free at the
other end
0.37
23
0.5
Figure 3.9: Flutter amplitude vs flutter frequency for different shell boundary conditions
0
0,5
1
1,5
2
2,5
0,95 0,96 0,97 0,98 0,99 1 1,01
clamped-clamped
simply supported-free
simply supported at both ends
Wnl/Wl
A/t
56
3.8.8 Effect of supersonic flow on the nonlinear behavior of cylindrical shells
The effect of supersonic flow on the nonlinear behavior of unpressurized cylindrical shells is
presented in Figure 3.10. The flutter frequency decreases rapidly at higher values of flutter
amplitude for the shells subjected to supersonic flow than empty shells indicating the softening
type of nonlinearity becomes more severe. The results were found for the same data of the
cylindrical shells, therefore the existence of the airflow equation in linear form is capable to
influence the geometrically nonlinear cylindrical shell, which is refer to flow
velocity (supersonic), not to density.
Figure 3.10: Effect of supersonic flow in nonlinear behavior of cylindrical shells
M=3, p∞=0.5 psi, pm=0.0 psi, n=23.
3.8.9 Effect of circumferential wave number
In Figure 3.11, the effect of large amplitude on the flutter frequency for axial mode m = 1 and
various circumferential modes n (20 to 25) is presented. The closed cylindrical shell is simply
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
2,2
0,5 0,7 0,9 1,1
no flow fluid
with supersonic flow
Wnl/Wl
A/t
57
supported and has the same data as other cases studied. The Figure shows that the nonlinearity is
of the softening type for all circumferential modes. It is also seen that the flutter amplitude vs
flutter frequency becomes more pronounced as the circumferential wave number increases.
Figure 3.11: Flutter amplitude versus frequency flutter of cylindrical
shells at different circumferential wave numbers
3.9 Conclusion
The method developed in this paper demonstrates the combined effect of the nonlinearity
associated with the wall of a shell and linear supersonic fluid flow on the resulting flutter
frequency and flutter amplitude for the specific case of closed cylindrical shells subjected to
external supersonic flow. It is a hybrid method, based on Sanders-Koiter nonlinear thin shell
theory, the linear aerodynamic piston theory and the finite element method.
The method combines the advantages of finite element analysis which deals with complex shells,
and the precision of a formulation employing displacement functions derived from shell and fluid
theories. This approach results in fast and efficient convergence of the calculations. The
calculated values show good agreement with experimental and theoretical results published in the
0
0,5
1
1,5
2
2,5
0 0,2 0,4 0,6 0,8 1 1,2
n=20
n=21
n=22
n=23
n=24
n=25
Wnl/Wl
A/t
58
literature. These numerical results provide the nonlinear behaviors of cylindrical shells for
different boundary conditions when the shells are subjected to supersonic flow beyond the onset
of flutter. These results cannot be found using linear analysis. Several parameters which are
dependent on shell geometry and fluid flow are studied. In all study cases, the nonlinear softening
type is dominant, and mild flutter instability has been detected. The effect of geometric
nonlinearities associated with the walls is non-negligible and should be taken into account when
calculating the nonlinear flutter of a shell-flowing fluid system when the amplitude of vibration is
larger than the thickness of the shell.
59
3.10 Appendix
Sander’s linear equations for thin cylindrical shells in terms of axial, tangential and
circumferential displacements are:
The coefficients of characteristic Equation (5) are given by:
2928 11 44 14
22 2 2 2
26 9 1 44 11 45 11 66 5 14 7 11 44 14 11 11 3 44 3 11 14
9 11 24 14 12
42
24 1 7 44 9 11 55 45 66 1 9 7 11 3 25 55 3 14 11 11
2 4 2 2
2
12 4 . 2 2
ff p p p
r
nf f f p p p p p f rp f p p p r f p f p rf f pr
f p p p pr
nf f f p f p p p p f f f p f p p f p f p rrr
2 2
11 3 5 1 11 5 7 14 5 9 25 22 3 14 11 11
2 2
12 5 9 7 14 3 11 24 3 1 9 7 11 9 11 25 9 11 22 12
6 2 2 2 2
2 7 45 66 55 1 9 7 11 3 5 7
25 55 1 11
(2 ) (2 ) ( ). 2
2 2 2
2 4
1 . 2 2
f r f f f f rf f p rf f n r p rp f r p f p
p f f r f p f f r p f f f f p f p p f p p p
f n r f p p p f f f p f r f f
p r p rf f rf
3 5 11 25 11 55
4 2
1 7 24 25 1 9 7 11 3 12 5 7 3 25 3 55
2
25 22 1 11 11 25 11 55 3 5
2 2 2
22 1 9 7 11 3 25 22 11 25 11 22 3 12 7 12
4
0 1 7 22
1
2 2 2
2 1 1
1 1 2
2
f p p r p p
n r f f p p f f f p f p rf f f p f r p
p rp f f r p p r p p f f
n p f f f p f r p rp r p p p p f p f p
f n f f p r n
2
2 4 2 2 3
25 55 1 25 55 25 221p n r p n f n r p r p n r p rp
Where
2 2 2 3 3 2 2 2
15 33 63121 11 142 2 3 2 2 2 2
3 2 2
36 66
2 3 2 2
2
5121 22 2
1( , , , ) ( ) ( ) ( )( )
2
3 1( )( 2 )
2 2 2
1( , , , ) ( ) (
ij
ij
P P PPU U V W W W V V UL U V W P P P
x R x x x x R x x R R x R
P P W V U
R R x x R
PP PUL U V W P
R R x R
2 3 3 2
52 54 25 552 24
2 2 2 2 2 2 3 2
2 2 3 2 2
63 6633 362 2 2
3 3 2
423 41 43 2 2
1)( ) ( ) ( )( )
3 31 3 1( )( ) ( )( 2 )
2 2 2 2
( , , , ) ( )ij
P P P PPV W W W V
R R R R x R R R
P PV U W V UP P
R x R x R R x x R x
PU V WL U V W P P P
x R x x
4 4 3 3 3
45 634 4 2 2 2 2 2 2
4 3 3 3 3 2 4 3
66 51 52 55
2 2 2 2 2 2 2 3 3 2 4 4 3
4
5421 22
2 2 2 2
2 1( ) ( )
2 3 1( 2 ) ( ) ( )
2 2
( )
P PW W V U V
x R x x R R x x
P P P PW V U U V W W V
R x x R x R x R R
PP PU W VW
R x R x R
2 2
2524
2 2 3 2( )
PP W W V
R R
60
2
1 33 36 66
2
3 12 33 15 36 66
5 15 36 66
2
7 22 55 25
2
9 33 36 66
11 36 24 66 54
4
3 4
2
2
3 9 4
2 3 /
f p p R p R
f p p p p R p R
f p p p R R
f p p R p R
f p p R p R
f p p p R p R R
Matrices 33][ T , 83][ R , and 88][ A are defined as:
/ / /
/
/ / /
cos( ) 0 0
[ ] 0 cos( ) 0
0 0 sin( )
(1, ) , (2, ) , (3, ) 1,2,...,8
(1, ) , (2, ) 1, (3, ) , (4, ) , (5, )
(6, ) , (7, ) , (8, )
i i i
i
i i i
x R x R x R
i i
l Rii i i
l R l R l Rii
n
T n
n
R i e R i e R i e i
A i A i A i A i A i eR
A i e A i e A i e iR
1,2,...8
Expressions L and NL for cylindrical shells are given by:
2
2
2
2 2
2
2
1
1
1
2 3 1
2 2
L
U
x
VW
R
V U
x R
W
x
W V
R
W V U
R x R x R
and;
61
2 2
2 2
2
1 1 1
2 8
1 1 1
2 8
1
2
0
0
0
NL
W V U
x x R
W V UV
R x R
W W WV
R x x
Matrices pqB and
pqC
/p q x R
pq pqB b e
/p q x R
pq pqC c e
where
1 1 2
2
1 1sin
2 4pq pq pqb a b n
R
1 2
2
1sin cos
4pq pq pqc c c n n
R
with
1
pq p p p q q qa n n
1
pq p qb n n , 1,...,8p q
1
pq p qc n
Matrix 3 8fR
is defined by
0...0
...
0...0
][)/()/( 81 RxiRxi
f eeR
62
3.11 References
[1] Librescu L. Aeroelastic stability of orthotropic heterogeneous thin panels in the vicinity of the
flutter critical boundary, Part I: Simply supported panels. J. Mec 1965; 4: 51-76.
[2] Librescu L . Aeroelastic stability of orthotropic heterogeneous thin panels in the vicinity of
the flutter critical boundary, Part II: J. Mec 1967; 6:133-152.
[3] Olson MD, Fung YC. Comparing theory and experimental for the supersonic flutter of
circular cylindrical shells. AIAA 1967; 5:1849-1856.
[4] Evensen DA, Olson MD. Nonlinear flutter of a circular cylindrical shell in supersonic flow.
NASA 1967; TN D-4265.
[5] Evensen DA, Olson MD. Circumferentially travelling wave flutter of a circular cylindrical
shell. AIAA 1968; 6: 1522-1527.
[6] Carter LL, Stearman RO. Some aspects of cylindrical shell panel flutter. AIAA 1968; 6: 37-
43.
[7] Barr GW, Stearman RO, Aeroelastic stability characteristics of cylindrical shells considering
imperfections and edge constraint. AIAA 1969; 7:912-919.
[8] Barr GW, Stearman RO. Influence of a supersonic flow-field on the elastic stability of
cylindrical shells, AIAA 1970; 8: 993-1000.
[9] Vol’mir AS, Medvedeva, Investigation of the flutter of cylindrical panels in a supersonic gas
flow. Sov Phys Dokl 1973; 17: 1213-1214.
[10] Amabili, M. Pellicano, (2001), Nonlinear supersonic flutter of circular cylindrical shells,
AIAA J.39, 564-573.
[11] Horn, W, et al. Recent Contributions to Experiments on Cylindrical Shell Panel Flutter.
AIAA Journal 1974; 12(11):1481-1490.
[12] Olson MD, Fung YC. Supersonic flutter of circular cylindrical shells subjected to internal
pressure and axial compression. AIAA 1966; 4: 858-864.
[13] Radwan H, Genin J. Non-linear modal equations for thin elastic shells. International Journal
of Non-linear Mechanics 1975; 10:15-29.
[14] Sanders, J.L. An Improved First-Approximation Theory for Thin Shell. NASA 1959; R-24.
[15] Lakis, A.A, M.P. Paidoussis. Dynamic Analysis of Axially Non-Uniform Thin Cylindrical
Shells. Journal of Mechanical Engineering Science 1972; 14(1):49-71.
63
[16] H. Ashley and G. Zartarian, Piston Theory-New Aerodynamic Tool for Aeroelastician,
Journal of the Aeronautical Sciences, 23 (1956),1109-1118.
[17] W. Horn et al, Recent Contributions to Experiments on Cylindrical Shell Panel Flutter,
AIAA Journal, 12 (1974), 1481-1490.
[18] E.H. Dowell, Flutter of Infinitely Long Plates and Shells, Part I, AIAA Journal, 4(1966),
1510-1518.
[19] E.H. Dowell, Aeroelasticity of Plates and Shells, Noordhoff International Publishing,
Leyden, 1975.
[20] A. Selmane, and A.A. lakis, Non-Linear Dynamic Analysis of orthotropic Open Cylindrical
Shells Subjected to a Flowing Fluid, Journal of Sound and Vibration, 202(1997), 67-93.
[21] M. Amabili, F. Pellicano, M.P. Paidoussis, Non-linear vibrations of simply supported circular
cylindrical shells coupled to quiescent fluid, J. Fluids Struct. 12(1998), 883-918
64
CHAPITRE 4: ARTICLE 3 - EFFECT OF GEOMETRIC AND FLUID
FLOW NONLINEARITIES ON CYLINDRICAL SHELLS SUBJECTED TO
A SUPERSONIQUE FLOW
Submitted, 07 June 2012 to International Journal of Numerical Methods and Applications
Redouane Ramzi and Aouni Lakis
4.1 Abstract
An analytical model incorporating a combination of the finite element method, Sanders-Koiter
nonlinear shell theory and third-order piston theory is presented in this paper to study the
influence of nonlinearities associated with structure geometry and supersonic fluid flow on the
dynamic and flutter instability behavior of thin anisotropic cylindrical shells. The shell is
subdivided into cylindrical finite elements and the displacement functions are derived from exact
solutions of Sander’s equations for thin cylindrical shells. These functions are treated using the
finite element method, which is developed to determine the linear and nonlinear mass and
stiffness matrices of the structure. The internal and external pressure and axial compression are
taken into account. All expressions of founding matrices are determined by exact analytical
integration. By applying the nonlinear dynamic pressure, which represents the effect of
nonlinearities of supersonic fluid flow on cylindrical shells, the linear and nonlinear matrices for
stiffness, damping and coupling are found. The nonlinear equation of motion is then solved using
a fourth-order Runge-Kutta numerical method. Nonlinear vibrations are determined with respect
to the amplitude of vibrations and thickness ratio for different cases. This approach leads to the
development of a powerful model capable of predicting linear, nonlinear vibrations and flutter
instability of cylindrical shells subjected to external supersonic flow. One specific application of
this model is the aeroelastic design of aerospace vehicles. This research is part of a series of
current studies in the literature that tackle the effects of the nonlinearity of external supersonic
fluid flow on cylindrical shells; an area of study that has been neglected by most research to-date.
4.2 Nomenclature
a∞ Freestream speed of sound
Ai Motion amplitude
65
[A] Matrix defined in Appendix
[B] Defined by Equation 11
[Cf] Aerodynamic damping matrix of fluid
[CFNL2
] Second order aerodynamic damping matrix of fluid
[CfNL3
] Third order aerodynamic damping matrix of fluid
ap,bp,cp Modal coefficients determined by equation 20
apq,bpq,cpq Coefficients determined by equations in Appendix
a(i)
pq,b(i)
pq,c(i)
pq Coefficients determined by equations in Appendix
Apq, Bpq, Cpq Modal coefficients determined by equations in Appendix
Aprs, Bprs, Cprs Modal coefficients determined by equation 19
Dprs, Eprs
Aprsq, Bprsq, Cprsq Modal coefficients determined by equation (19, 22)
Dprsq, Eprsq
D Membrane stiffness, defined by Equation 3
{ɛ} Deformation vector giving in Appendix
E Young’s modulus
[E] Matrix defined by Equation 23
{Fp} Force vector due to the aerodynamic pressure field
h Shell thickness
K Bending stiffness of shell, defined by Equation 3
[Ks] Global stiffness matrix for a shell
[k] Stiffness matrix for a shell element
[kf] Local aerodynamic stiffness matrix
[kI] Initial stiffness matrix for a shell element
[Kf] Global aerodynamic damping matrix
[KI] Global initial stiffness matrix for a shell
[KNL2
] Nonlinear stiffness matrix of second order
[KNL3
] Nonlinear stiffness matrix of third order
L Shell length
M Mach number
[Ms] Global mass matrix for a shell
66
n Circumferential number
[m] Mass matrix for a shell element
N Cylindrical shell number
Nx,Nθ,Nxθ Stress resultant for a circular cylindrical shell
xN ,N Stress resultant due to shell internal pressure and axial compression
[N] Defined by Equation 10
pa Aerodynamic pressure
p∞ Freestream static pressure
pm, px Shell internal pressure and axial compression
[P] Elasticity matrix
Qx, Qθ Transverse stress resultant for a circular cylindrical shell
R Shell radius
[Rf] Matrix defined in Appendix
U Axial displacement
Ui Potential energy due to initial strain
V Circumferential displacement
W Radial displacement
x Longitudinal coordinate
δi, δj Displacement at node i and j
λj Complex roots of the characteristic equation
υ Poisson’s ratio
γ Adiabatic exponent
θ Circumferential coordinate
ρ Shell density
ρf Fluid density
ω Oscillation frequency
øxx, øxθ, øn Elastic rotations for a circular cylindrical shell
{σ} Stress vector
67
4.3 Introduction
The design of modern missiles, military fighter planes, space shuttles and aerospace structures
requires a very advanced, precise and high-performance analytical method capable of
investigating the effects of various parameters that influence the reliability and control of
sensitive functional and safety systems.
These structures are composed of elements of shells and plates subjected to the effects of external
supersonic and hypersonic airflow, which create a phenomenon of dynamic instability.
Cylindrical shells also exhibit this type of aeroelastic instability, and prevention of this behavior
has become a priority in the aerospace and military industry.
The V-2 rocket was the first cylindrical shell known to have experienced the phenomenon of
flutter instability. From the moment the phenomenon was discovered, the study of dynamic
instability and flutter became crucial, especially after the introduction of Piston theory published
by Ashley and Zartarian [1]. Many authors have investigated the stability of shells in supersonic
flow using a linear model of shell and fluid flow through both theoretical and experimental
studies. Most researchers have tried to find an analytical formulation assembling various
parameters of the shell and the fluid flow to develop a predictive model of aeroelastic flutter
characteristics that may describe the reaction of cylindrical shells. Their models were based on
shell theory (linear and nonlinear Donnell thin shell theory) combined with linear and nonlinear
modeling using the piston theory for fluid-structure interaction. The equation of motion is solved
using the Galerkin method. However, the results obtained are not in agreement with those
observed experimentally [2]. After Olson and Fung [3] studied the boundary conditions and the
effects of internal pressure and axial compression, they concluded that cylindrical shells flutter
due to a low-level freestream static pressure as cited by the theory [4]. A nonlinear study to
calculate the limit cycle amplitude of cylindrical shells using four modes for the deflection of the
shell is given by Evensen and Olson [5]. They reached the same result as given in Ref [3],
indicating that flutter is a type of circumferential travelling wave. Also Dowell [6,7] analyzed the
behaviour of cylindrical shells in supersonic flow by studying various parameters of the shell and
flow. Barr and Stearman [8,9] considered the initial geometric imperfections in cylindrical shells.
They noticed that supersonic flow has no effect on the critical buckling load when a small
amount of axial load decreases the speed of the flutter. Amabili and Pellicano [10] studied the
68
geometric nonlinearities of cylindrical shells in a supersonic regime based on the selection of
expansion modes. In their study [11] they apply pressure derived from nonlinear piston theory on
shells with imperfections in order to study fluid-structure interaction. They conclude that the
nonlinearity has no influence on the onset of flutter. In the literature there are several studies on
the dynamic behavior and instability of cylindrical shell flutter but these are generally either
based on linear analysis of shell and fluid flow, or nonlinearity of the shell and linearity of the
fluid flow, or a condition where both are nonlinear. The case which deals with linear analysis of a
structure subjected to nonlinear fluid flow has not yet been investigated. This current study
consists of an in-depth investigation of the nonlinear dynamic behaviour of structures and their
stability versus the nonlinearity of the fluid flow. It has helped fill the void in this specific area
and advanced the field of design and preventive control of cylindrical shell structures used under
supersonic conditions, which are often found in the aerospace industry. The structural modeling
approach using the finite element method for cylindrical shells is inspired from a Lakis and
Paidoussis study [12]. This methodology is characterized by its accuracy and rapidity with less
numerical difficulties because the displacement functions are derived directly from equations
governing the shell instead of polynomial functions. This study presents the effects of the
nonlinearities of geometry and fluid flow on the dynamic behavior of cylindrical shells under
supersonic airflow. The method is based on a combination of the finite element method, Sanders-
Koiter nonlinear theory [11, 12, 13, 16, 17, 18, 19 and 20] and nonlinear aerodynamics theory.
This method satisfies the finite element method convergence criteria [12] and provides greater
accuracy than polynomial functions, which are typically chosen for this type of analysis.
The treatment of this problem involves two steps:
1. Using linear strain-displacement, stress-strain relationships which were introduced into
Sander’s equation of equilibrium, the modal displacements describing large amplitude
vibration are derived using the developed method seen in the Radwan and Genin study
[21]. Then all the displacement functions are determined by solving the system of
equations. Following this, the linear and nonlinear mass and stiffness matrices for the
finite element are found and assembled for the complete shell.
2. From third-order piston theory [1, 22] we derive an expression for nonlinear pressure as
a function of:
69
- the nodal displacement of the shell element
- a combination of nonlinear effects.
Using this nonlinear dynamic pressure, we obtain two linear matrices and four nonlinear matrices
from third-order piston theory.
4.4 Structural model development
4.4.1 Masse and stiffness matrices for linear structural model
The equations of motion of an anisotropic cylindrical shell in terms of U, V, and W (axial,
tangential and radial displacements) (Figure 4.1) are written as follows:
0),,,( ijJ PVWUL
Where ( 1,2,3)J JL are three linear partial differential equations (given in Appendix) and ijp are
elements of the elasticity matrix for anisotropic shells which is given by:
11 12 14 15
21 22 24 25
33 36
41 42 44 45
51 52 54 55
63 66
0 0
0 0
0 0 0 0[ ]
0 0
0 0
0 0 0 0
p p p p
p p p p
p pP
p p p p
p p p p
p p
For the isotropic case,
0 0 0 0
0 0 0 0
(1 )0 0 0 0 0
2[ ]
0 0 0 0
0 0 0 0
(1 )0 0 0 0 0
2
D D
D D
D
PK K
K K
K
where,
3
212(1 )
EhK
Bending stiffness,
21
EhD
Membrane stiffness
The displacement functions associated with the circumferential wave number are assumed in the
normal manner, as:
(2)
(1)
(3)
70
( , ) cos
( , ) cos
( , ) sin
x
R
x
R
x
R
U x Ae n
W x Be n
V x Ce n
Figure 4.1: Finite element discretization and nodal displacement
where n is the circumferential mode,
Note that this approach treats closed cylindrical shells based on the circumferential wave number,
not on the longitudinal wave number.
Substituting (4) into the equations of motion (1), a system of three homogeneous equations, linear
functions of constants A, B, C, are obtained. For the solution to be non-trivial, the determinant of
this system must be equal to zero. This brings us to the following characteristic equation:
8 6 4 2
8 6 4 2 0Det H f f f f f
The values of coefficients if in this eighth-degree polynomial are given in Appendix.
Each root of this equation yields a solution to the linear equations of motion (1). The complete
solution is obtained by adding the eight solutions independently with constants Ai, Bi and Ci
(i = 1,2,….,8). These constants are not independent. We can therefore express Ai and Bi as a
function of Ci, as follows:
i i i
i i i
A C
B C
(4)
W
U
W
V
un
dwn/dx
wn
xj=l
xi=0
(5)
(6)
71
The values of i and i can be obtained from system (1) by introducing relation (6). To
determine eight iC constants, it is necessary to formulate eight boundary conditions for the finite
elements. The axial, tangential and radial displacements will be specified for each node. The
following relation defines the element displacement vector at the boundaries:
{ } { }
CAj
i
where the elements of matrix [A] are given in Appendix.
Multiplying Eq.(8) by [A]-1
and substituting the result into Eq.(4), we obtain:
j
iN
,xW
,xV
,xU
where [N] represents the shape function matrix given in this form:
and [T], [R], [A] are described in Appendix.
The constitutive relation between the stress and deformation vector of a cylindrical shell is given
as:
j
iBP
]][[
The mass and stiffness matrices for each element are derived using the classic finite element
procedure and can then be expressed as:
(11)
1N T R A
(10)
(7)
(8)
(9)
72
dA]B][P[]B[]k[
dA]N[]N[h]m[
TL
T
Where is the density of the shell, h its thickness and dA Rdxd . The matrices [m] and [k] are
obtained analytically by carrying out the necessary matrix operations over x and θ in the
equation. The global matrices sM and sK may be obtained, respectively, by assembling the
mass and stiffness matrices for each individual finite element.
4.4.2 Stiffness matrices for nonlinear structural model
The coefficients of the modal equations are obtained using the Lagrange method in the same way
developed by Radwan and Genin [21], paying attention to geometric non-linearities. Following
this, the second and third non-linear stiffness matrices corresponding to geometric non-linearities
are computed and superimposed on the linear system.
Shell displacements (u, v, w) are expressed as sums of generalized product coordinates iq t and
spatial functions U, V, W (see Eq. (4)).
,
,
,
i i
i
i i
i
i i
i
u q t U x
v q t V x
w q t W x
The deformation vector is written as a function of the generalized coordinates by separating the
linear part from the non-linear part.
, ,2 , , ,2T
L NL xx x xx x
Where subscripts ``L`` and ``NL`` mean ``linear`` and ``non-linear``, respectively, and L and
NL are defined in Appendix.
(12)
(13)
(14)
73
The Lagrange’s equations of motion in generalized coordinates iq t are found using Hamilton’s
principle and Eq. (13):
iiii
V
q
T
q
T
dt
d
where T is the total kinetic energy, V is the total elastic strain energy of deformation and the iQ ’s
are the generalized forces.
Using the vector deformation given by (14), the developed formulation for total kinetic and strain
energy and inserting these into the Lagrange equation (15), the following non-linear modal
equations are obtained after a long development of equations:
( ) ( 2) ( 3) 0
1,2,....
L NL NL
pr r pr r prs r s prsq r s q
r r r s r s q
m k k k
p
where prm , L
prk are terms of the mass and linear stiffness matrices given by Equation (12).
The terms )NL(
prsk 2 and
)NL(prsqk
3 represent the second and third-order non-linear stiffness and
are given by the following integrals for the case of an isotropic closed cylindrical shell:
dACp)ED(pBpApK prsprsprsprsprs)NL(
prs 331222112
and
( 3)
11 22 12 33( )NL
prsq prsq prsq prsq prsq prsqK p A p B p D E p C dA
Where dA Rdxd , ijp are the terms of the elasticity matrix [P], and the terms
Aprs,Bprs,Cprs,Dprs,Eprs and Aprsq,Bprsq,Cprsq,Dprsq,Eprsq represent the coefficients of the modal
equations. These coefficients are given by the following equations:
(15)
(16)
(17)
(18)
74
Aprs=ap Ars+ar Asp+as Apr Aprsq=2Apq Ars
Bprs=bp Brs+br Asp+bs Bpr Bprsq=2Bpq Brs
Cprs=cp Crs+cr Csp+cs Cpr Cprsq=2Cpq Crs
Dprs=ar Bsp+as Bpr+bp Ars Dprsq=2Apq Brs
Eprs=br Asp+bs Apr+as Brs Eprsq=2Bpq Ars
with;
ap = Up,x , bp= 1/R ( Vp,θ + Wp ), cp= 1/2 ( Up,θ / R + Vp,x )
Apq =1/8 R2 ( R Vp,x – Up,θ ) . ( R Vq,x – Uq,θ ) + 1/2 Wp,xWq,x
Bpq = 1/8 R2 ( R Vp,x – Up,θ ) . ( R Vq,x –Uq,θ ) + 1/2 R
2 ( Wp,θ - Vp ) . ( Wq,θ - Vq )
Cpq = 1/4 R ( Wp,x Wq,θ – Wq,x Wp,θ ) – 1/4 R ( Vp Wq,x + Vq Wp,x )
Where U, V and W are spatial functions determined by Equation (4).
In Equations (16) - (20), the subscripts ‘p,q’, ’p,r,s’ and ‘p,r,s,q’ represent the coupling
between two, three and four modes, respectively.
Let us illustrate the development of the expression for the (p,q) term of matrices Apq and Apqrs.
The expression Apq is given by:
/p q x R
pq pqA a e
and Apqrs is:
8 8( ) /
1 1
p r s q x R
prsq rs pq rs
r s
A a a e
rs is the term (r,s) of matrix [E],where [E] represents a matrix of constants defined by
1T
E A A , and
(19)
(20)
(21)
(22)
(23)
75
1 22 2
2
1 1sin cos
2 4pq pq pqa a n a n
R
with
1
pq p p p q q qa n n
2
pq p qa
Similarly, the matrices [Bpq]… [Eprsq] can be written as a function of α, β, λ, x and θ. The (p, q)
terms of these matrices are described in Appendix. For details of the basis of the presented
method, consult reference [21].
After integration, we obtain:
2 21 1TNL NL
prsk A J A
3 31 1
28
TNL NL
prsqk A J AR
The (p,q) term in matrix 2
,
NL
p qJ
and 3
,
NL
p qJ
are written as:
8 8
1 1
2
8 8
1 1
1 , 1
0,
1 ,
0
p q rsr R
l s p q r
NL p q r
sr
k l
p q r
G p q e
ifJ p q
G p qR
if
and
(24)
(25)
(26)
(28)
(29)
(27)
76
8 8
1 1
3
8 8
1 1
2 , 1
0,
2 ,
0
p q r ssr R
k l p q r s
NL p q r s
sr
k l
p q r s
G p q e
ifJ p q
G p qR
if
where G1(p,q) and G2(p,q) are coefficients in conjunction with α, β, λ and elements pij in matrix
[P]. The general expressions of G1(p,q) and G2(p,q) are given by :
1 1 1 1 1 11 1 1
11 1
1 2 1 2 1 21 1 1
11 2
1 2 1 2 1 21 1 1
22 1
1 1 1 1 1 11 1 1
33 1
1 11
12 1
G1(p,q) = p rs r sp s pr
p rs r sp s pr
p rs r sp s pr
p rs r sp s pr
r sp
p II a A a a A a a A a
p II a A a a A a a A a
p II b A b b A b b A b
p II c A c c A c c A c
p II a A b
1 1 1 11 1
1 21
12 2
1 1 1 1 1 11 1 1
12 1
1 2 1 21 1
12 2
s pr p rs
p rs
r sp s pr p rs
r sp s pr
a A a b A a
p II b A a
p II b A a b A a a A b
p II b A a b A a
where,
3
1
sin 2
3
nII
2
2
sin(2 )(2 cos (2 ))
3
n nII
n
and
(30)
(31)
(32)
(33)
77
1 1 2 2 1 1
11 22 12 11 22
2 1 1 2 2 1 1 2
12 11 12
1 1 1 1
12 22
1 1 2 2 2 1 1 2
33
32( , ) 2 3
16
1
4
3
4
1
4
pl kq pl kq pl kq
pl kq pl kq pl kq pl kq
pl kq pl kq
pl kq pl kq pl kq pl kq
G p q p p p a a p a a p b b
p a b b a p p a a a a
p p a b b a
p c c c c c c c c
where the terms 1
a and
2a are given by Equations (25-26). Terms b
(1)…, c
(1)…and c
(2) are
coefficients appearing in expressions for the elements of matrices [Bpq] and [Cpq] defined in
Equations (20). These coefficients are given in Appendix.
4.5 Initial stress matrix
The potential energy is given as
dxRdNNNNUl
nxxxxi
0
2
0
222 )(2/1
where;
RpN
R
pN
m
xx
2
xN , N are stress resultants due to shell internal pressure mp and axial compression xp
respectively, l is the element length, is about the normal to the x plane, xx is the strain
rotation about x , and n is the rotation about the normal to a shell element.
This rotation vector is expressed as [14],
)1
(2
1
)(1
x
VU
R
WV
R
x
W
n
xx
(34)
(35)
(37)
(36)
78
Upon insertion of displacements (U,W,V) given in Equation (8), the potential energy is rewritten
in terms of the nodal degrees of freedom, which leads to the initial stiffness matrix for each
element in the following form:
dxRdNC
NN
N
N
CNkl
TT
x
x
TT
I
0
2
000 ][][
00
00
00
][][][
where;
xR
R
x
C
2
10
2
1
01
0
00
][ 0
4.6 Aerodynamic model development
The pressure on the body surface is approximated by the pressure experienced by a piston
moving in the flow with a velocity equivalent to the superposition of the velocity due to the
changing of body shape and the velocity due to the rigid-body motion of the body [1, 22].
12
211
/
a/
p
p
where;
p is the aerodynamic pressure,
p∞ is the freestream static pressure,
γ is the adiabatic exponent,
a∞ is the freestream speed of sound
ω is the velocity of the piston.
The piston velocity ω is given by:
(40)
(39)
(38)
79
x
WU
t
W
is the freestream velocity and W is the radial displacement.
Considering the linear terms in the binomial expansion of Equation (40), we obtain:
2
3
1 1 1
4
1 1
12
a
W W W WM M
x a t x a tp p
W WM
x a t
The form of the radial displacement assumed is the same as that considered by Lakis and
Paidoussis in [12]:
8
1
8
1 j
j
j
)tR
x(i
W)ncos(eWj
where; j is the j
th root of the characteristic equation, is the natural angular frequency and n
is the circumferential wave number.
Substituting Eq. (43) into Eq. (42), we obtain the equation for the aerodynamic pressure on the
cylindrical shell wall:
333
1 1 3 1
2223
1
2
1 1
12
1
4
i i ij j
a f f f
j jj
i ij
f
j j
i ip p M T R A T R A M T R A
R a R
MT R A
a R
2 3
3 31 1
2 3
1 1
4 12
i i ij
f f
j j j
iMT R A T R A
Ra a
Then the damping and stiffness for fluid flow are obtained by substituting Equations (9) and (44)
into the integral (45),
[ ] { }T
p aF N p dA
The linear matrix of aerodynamic damping [Cf] and stiffness matrix [Kf] are given as follows
(42)
(45)
(43)
U
(41)
(44)
80
1 1T
f fC A D A 1 1
33
T
f fnlNLC A D A
1 1
T
f fK A G A 1 1
33
T
f fnlNLK A G A
2 1 1
331
T
f f fnlNLK C A H A
2 1 1
332
T
f f fnlNLK C A H A
where ,
dxRRrPa
Dl
fT
f
0
dxRRrMPr
iG f
lTj
f
0
where fD , fG , 3fnlD , 3fnlG , 31fnlH , 32 fnlH are given in Appendix.
4.7 Nonlinear numerical analysis
The motion of a cylindrical shell subjected to an external supersonic flow is governed by the
equation of motion which may be presented as follows:
2 3 3 2
3 3 2( 2)
2
i i i i i i i iL L L NL NL NL NLNL
s f f f f
j j j j jj j j
NL NL
f f
M C K K K K K C
K C
2 3
30
i i iNL
f
j j j
C
where subscripts s and f refer to the shell in vacuo and in fluid flow respectively, and is the
degrees of freedom vector for the total nodes.
Further useful expressions are:
L L
s I fK K K K
(50)
(51)
(48)
(49)
(47)
(46)
81
dxRdNC
NN
N
N
CNK TT
x
xl
TTI
0
0
2
0
0
00
00
00
T
sK B P B dA
1 1T
L
f fK A G A
where
T Tk
f f
iG p M R T T R dA
R
Let us set:
q
where represents the square matrix for eigenvectors of the linear system and q is a time-
related vector. Substituting Equation (55) into system (50) and pre-multiplying the whole
equation by T
, we obtain;
2 32 3
3 23 2
2 32 30
D D D T TL L L NL NL
s f
T TNL NL
f f f
T TNL NL
f f f
M q C q K q K q K q
K q K C q q
K C q q C q
where;
D TL L
s sM M
, D TL L
f fC C
, D TL L
K K
D denotes diagonal, and the matrices quantifying the fluid contribution to the matrix equations of
motion are non-symmetric. To facilitate the analysis, therefore, we consider only the symmetric
portion of the matrices. This simplifying hypothesis is valid since the original and simplified
systems have comparable dynamic behavior.
(52)
(54)
(55)
(56)
(57)
(53)
82
By neglecting the cross-product in 2
q …of Equation (56), we obtain;
2 3 3 2 2 32 3 3 2 2 3
1
0NREDUC
Lf L NL NL NLf NLf NLf NLf
ii i ii i ii i ij j ij j ij j ii ij j j ii ij j j ij j
j
m q c q k q K q K q K q K C q q K C q q C q
Here we have ‘‘NREDUC’’ simultaneous equations of the form of Equation (58).
At first, we limit ourselves to solving Equation (56) by taking into account only the diagonal
terms of the products 22T NL
K
and 33T NL
K
, therefore Equation (58) is written
in this form:
2 3 3 2 2 32 3 3 2 2 3 0L L NL NL NLf NLf NLf NLf
ii i ii i ii i ii i ii i ii i ii ii i i ii ii i i ii im q c q k q K q K q K q K C q q K C q q C q
Setting
i i iq A f
with 10 if and 00 if
Equation (59) becomes, after the iA simplification and dividing by iim ,
2 2
2 2 3 3
2 2
2 2 2
2 2 3
2 2 20
i i i
ii i i i i i i i i i i
i i i
i i i i i i i i
A A Af f f f f f
t t t
A A Af f f f f
t t t
where;
ii
L
iii
m
c ;
ii
L
iii
m
k2 ;
2NL
ii
i
ii
Kt
m ;
3
2
NL
ii
i
ii
Kt
m ;
3
2
NLf
ii
i
ii
Kt
m ;
2
2
NLf
ii ii
i
ii
K Ct
m ;
2
2
NLf
ii ii
i
ii
K Ct
m ;
3
2
NL
ii
i
ii
Ct
m
and t represents the shell thickness. The square root of coefficient /ii iik m represents the ith
linear
vibration frequency of system. The solution if of these ordinary nonlinear differential
equations which satisfies the initial conditions (60) is approximated by a fourth-order Runge-
Kutta numerical method. The linear and non-linear natural angular frequencies are evaluated
(61)
(59)
(60)
(58)
(62)
83
using a systematic search for the if roots as a function of time. The /NL L ratio is expressed
as a function of the non-dimensional ratio /iA t , where iA is the vibration amplitude.
4.8 Calculation and Discussion
The influence of nonlinearities associated with the wall of the shell and with supersonic fluid
flow on closed cylindrical shell vibrations is expressed by Equation (59). For a given set of shell
characteristics, we first present results for convergence of the model and the ratio Wnl/Wl of linear
and nonlinear frequency is graphically represented with respect to the non-dimensional ratio Ai/t.
The shell and flow properties are given by:
6 10 2
2 4 3
16 10 / (11 10 / )
0.35
0.0040 (0.0001015 )
15.4 (0.381 )
8.00 (0.203 )
0.000833 / (8900 / )
3.00
8400 / (213 / )
s
E lb in N m
h in m
L in m
R in m
lb s in kg m
M
a in s m s
4.8.1 Convergence of the method for different modes
A first calculation was focused on determining the required number of finite elements for a
precise determination of results, which is very important in finite element analysis. Calculations
were made for a closed circular cylindrical shell subjected to supersonic flow for the number of
finite elements N = 2, 4, 6, 8...25. The shell is simply supported at both ends. The results for n =
23 and for m = 1, 2 and 3, are shown in Figure 4.2. Convergence of natural frequency is reached
for 20 elements.
84
Figure 4.2: Naturel frequency vs Number of elements
4.8.2 Onset of flutter predictions for cylindrical shells subjected to a
supersonic flow
Figure 4.3 shows the eigenvalues associated with the longitudinal modes and n = 23 (critical
circumferential wave number) versus freestream static pressure p∞, knowing that the real part of
the eigenvalues gives the oscillation frequency and the imaginary part gives the damping for
positive values and instability for negative values. Flutter starts where the imaginary parts cross
the zero and become negative. It is seen that the shell loses stability at pcr = 0.75 psi, which is
called the critical freestream static pressure. This instability is known as coupled-mode flutter.
220
240
260
280
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
1st mode
2nd mode
3rd mode
Number of elements
Fre
qu
ency
(H
z)
85
Figure 4.3: a) Real part and b) Imaginary part of eigenvalues of the system
vs freestream static pressure p∞, n = 23, pm = 0.5 psi, px = 0.0 psi.
4.8.3 Validation of model
In Figure 4.4, validation of our model is shown based on the effects of both geometric
and supersonic flow nonlinearities which are modeled by a combination of the Koiter-
Sanders nonlinear theory for large amplitude vibrations of circular cylindrical shells and third-
order of piston theory for describing fluid-structure interaction in supersonic regime. This results
show good agreement between the study developed in this paper and Olson research [3]. These
results also show that the variation ratio of the nonlinear frequency and linear frequency with
respect to increasing of the mean amplitude of the first longitudinal mode (m=1) coincide for first
order piston theory (linear) and third order Piston theory (nonlinear). This reveals that the effect
of the nonlinearity of supersonic flow is negligible compared to the effect of geometric
nonlinearity of circular cylindrical shells subjected to supersonic flow. It is also noted
that the nonlinearity is always the softening type, a result which is in quite good agreement with
previously published work.
548
549
550
551
552
553
554
555
556
557
558
0 0,2 0,4 0,6 0,8 1 1,2
Freestream Static Pressure (psi)
Rea
l ei
gen
val
ue
(Hz)
-1
0
1
2
3
4
5
6
7
8
0 0,2 0,4 0,6 0,8 1
Freestream Static Pressure (psi) Im
agin
ary e
igen
val
ue
(Hz)
86
Figure 4.4: Flutter frequency vs amplitude, pec=0.66 psi.
4.8.4 Effect of circumferential wave mode
Two circumferential wave numbers n = 23 and n = 24 are considered in this section for studying
the effect of circumferential mode on the flutter frequency and flutter amplitude for the same
physical and mechanical data of the closed circular cylindrical shell described in the previous
sections. On one hand, Figure 4.5 shows a more pronounced variation of flutter amplitude versus
flutter frequency at n =24. On the other hand, no difference is observed between linear (1storder)
and nonlinear (3rd order) modeling for both circumferential modes in this study. The same
behavior is observed for other circumferential modes.
0
0,2
0,4
0,6
0,8
1
1,2
0,96 0,97 0,98 0,99 1 1,01
Olson
linear piston
nonlinear Piston
Nondimensional flutter frequency, Wnl/Wl
Mae
n a
mp
litu
de
of
firs
t m
od
e, A
87
Figure 4.5: Amplitude flutter vs frequency flutter at n=23 and n=24.
4.8.5 Effect of internal pressure
Circular cylindrical shells subjected to internal pressure are analyzed here. The goal is to
determine how nonlinear behavior changes the onset of flutter, taking into account the effect of
linear or nonlinear supersonic flow. Figure 4.6 shows that curves illustrating the response of
two linear and nonlinear approximations of the onset of flutter (characterized by pcr =0.75 psi )
coincide. This means that the effect of the nonlinearity of supersonic flow is still negligible at the
critical pressure value (at which the flutter begins).
It is also seen that, due to the internal pressure, the frequency ratio does not change as the
variation in flutter amplitude increases. However, at a larger value of fluid pressure
pec = 2.0 psi, variation of flutter amplitude is smaller, and the nonlinear frequency tends toward
the value of the linear frequency. This demonstrates the stabilizing effect of internal pressure
during the flutter of circular cylindrical shells. The same results were observed in my previous
studies on linear flutter in the supersonic regime and also in Amabili [10], and Olson and Fung
[3].
0
0,5
1
1,5
2
2,5
0,93 0,94 0,95 0,96 0,97 0,98 0,99 1 1,01
n=24 , linear Piston
n=24, nonlinear Piston
n=23, linear Piston
n=23, nonlinear Piston
Nondimensional flutter frequency, Wnl/Wl
Flu
tter
am
pli
tud
e, A
/t
88
Figure 4.6: Effect of internal stress during the flutter, n=25, pec =0.75 psi.
4.8.6 Effect of supersonic flow regimes in flutter of circular cylindrical shells
Figure 4.7 shows a comparison between unpressurized circular cylindrical shells under different
conditions: empty shells (no fluid flow), linear supersonic flow and nonlinear supersonic flow.
A gradual decrease of the flutter frequency as flutter amplitude increases is very evident in the
case of nonlinear supersonic flow compared to the other cases. The nonlinear effect of the
supersonic flow is almost constant (Wnl / Wl = 1). This expresses that the fluid plays its usual
role of vibration damping for different amplitudes of wall movement for the cylindrical panels.
0
0,5
1
1,5
2
2,5
0,92 0,94 0,96 0,98 1 1,02
1st order,pm=0,pec=0.75psi
3rd orde, pm=0.5psi
3rd order, pm=0
3rd order, pm=0, pec=2.0 psi
Nondimensional flutter frequency, Wnl/Wl
Flu
tter
am
pli
tud
e ,
A/t
89
Figure 4.7: Flutter amplitude vs frequency flutter for different L/R.
4.8.7 Nonlinear flutter behavior for different longitudinal modes
Figure 4.8 presents the behavior of nonlinear flutter for circular cylindrical shells subjected
to supersonic fluid flow at different longitudinal wave numbers m = 2, ..., 7. It is seen that each
curve corresponds to two longitudinal modes because the flutter phenomenon is created by the
coalescence of the real parts of successive complex frequencies of two modes. This was seen
in Figure 4.3a, and can be predicted by linear analysis of cylindrical shells in the supersonic
regime. It can also be predicted by nonlinear analysis using the model presented in this paper.
Furthermore the nonlinearities are of the softening type, and the trend of the curves is the same
for all three cases of flutter considered in this study, including a slightly
pronounced deviation for the case m = 2,3 at high values of vibration amplitude.
0
0,5
1
1,5
2
2,5
0 0,2 0,4 0,6 0,8 1 1,2
without flow
1st order Piston theory
3rd Piston theoryF
lutt
er a
mp
litu
de
, A
/t
Nondimensional flutter frequency, Wnl/Wl
90
Figure 4.8: Flutter amplitude vs flutter frequency for different longitudinal modes
4.9 Conclusion
The method developed in this paper demonstrates the influence of geometric nonlinearity
associated with the wall of the shell and nonlinear supersonic fluid flow (3rd
order of Piston
theory) on the frequency and amplitude of flutter of closed cylindrical shells subjected to external
supersonic flow. It is a hybrid method based on Sanders-Koiter nonlinear thin shell theory, and
nonlinear aerodynamic piston theory and the finite element method. The method combines the
advantages of finite element analysis which deals with complex shells, and the precision of a
formulation in which the use of displacement functions derived from shell and fluid theories
allows fast and efficient convergence of the calculations. The results show better agreement with
experimental and theoretical results published in the literature. The numerical results provide the
nonlinear behaviors for circular cylindrical shells subjected to supersonic flow beyond the onset
of flutter. These results cannot be determined by linear analysis. Several parameters that are
dependent on geometry and fluid flow are studied, and in all cases nonlinearity of the softening
type is dominant. Mild flutter instability and the stabilizing effect of internal pressure has been
0
0,2
0,4
0,6
0,8
1
1,2
0,975 0,98 0,985 0,99 0,995 1 1,005
m=2,3
m=6,7
m=4,5
Nondimensional flutter frequency, Wnl/Wl
Flu
tter
am
pli
tud
e ,
A/t
91
demonstrated. The effect of geometric nonlinearities associated with the walls is non-negligible
and should be taken into account when calculating the nonlinear flutter of a shell-fluid flow
system. In contrast the nonlinearity of supersonic fluid flow can be neglected when the amplitude
of vibration is larger than the thickness of the shell.
92
4.10 Appendix
Sander’s linear equations for thin cylindrical shells in terms of axial, tangential and
circumferential displacements are
- The coefficients of characteristic Equation (5) are given by
2928 11 44 14
22 2 2 2
26 9 1 44 11 45 11 66 5 14 7 11 44 14 11 11 3 44 3 11 14
9 11 24 14 12
42
24 1 7 44 9 11 55 45 66 1 9 7 11 3 25 55 3 14 11 11
2 4 2 2
2
12 4 . 2 2
ff p p p
r
nf f f p p p p p f rp f p p p r f p f p rf f pr
f p p p pr
nf f f p f p p p p f f f p f p p f p f p rrr
2 2
11 3 5 1 11 5 7 14 5 9 25 22 3 14 11 11
2 2
12 5 9 7 14 3 11 24 3 1 9 7 11 9 11 25 9 11 22 12
6 2 2 2 2
2 7 45 66 55 1 9 7 11 3 5 7
25 55 1 11
(2 ) (2 ) ( ). 2
2 2 2
2 4
1 . 2 2
f r f f f f rf f p rf f n r p rp f r p f p
p f f r f p f f r p f f f f p f p p f p p p
f n r f p p p f f f p f r f f
p r p rf f rf
3 5 11 25 11 55
4 2
1 7 24 25 1 9 7 11 3 12 5 7 3 25 3 55
2
25 22 1 11 11 25 11 55 3 5
2 2 2
22 1 9 7 11 3 25 22 11 25 11 22 3 12 7 12
4
0 1 7 22
1
2 2 2
2 1 1
1 1 2
2
f p p r p p
n r f f p p f f f p f p rf f f p f r p
p rp f f r p p r p p f f
n p f f f p f r p rp r p p p p f p f p
f n f f p r n
2
2 4 2 2 3
25 55 1 25 55 25 221p n r p n f n r p r p n r p rp
2 2 2 3 3 2 2 2
15 33 63121 11 142 2 3 2 2 2 2
3 2 2
36 66
2 3 2 2
2
5121 22 2
1( , , , ) ( ) ( ) ( )( )
2
3 1( )( 2 )
2 2 2
1( , , , ) ( ) (
ij
ij
P P PPU U V W W W V V UL U V W P P P
x R x x x x R x x R R x R
P P W V U
R R x x R
PP PUL U V W P
R R x R
2 3 3 2
52 54 25 552 24
2 2 2 2 2 2 3 2
2 2 3 2 2
63 6633 362 2 2
3 3 2
423 41 43 2 2
1)( ) ( ) ( )( )
3 31 3 1( )( ) ( )( 2 )
2 2 2 2
( , , , ) ( )ij
P P P PPV W W W V
R R R R x R R R
P PV U W V UP P
R x R x R R x x R x
PU V WL U V W P P P
x R x x
4 4 3 3 3
45 634 4 2 2 2 2 2 2
4 3 3 3 3 2 4 3
66 51 52 55
2 2 2 2 2 2 2 3 3 2 4 4 3
4
5421 22
2 2 2 2
2 1( ) ( )
2 3 1( 2 ) ( ) ( )
2 2
( )
P PW W V U V
x R x x R R x x
P P P PW V U U V W W V
R x x R x R x R R
PP PU W VW
R x R x R
2 2
2524
2 2 3 2( )
PP W W V
R R
93
where;
2
1 33 36 66
2
3 12 33 15 36 66
5 15 36 66
2
7 22 55 25
2
9 33 36 66
11 36 24 66 54
4
3 4
2
2
3 9 4
2 3 /
f p p R p R
f p p p p R p R
f p p p R R
f p p R p R
f p p R p R
f p p p R p R R
- Matrices 33][ T , 83][ R , and 88][ A are defined as
/ / /
/
/ / /
cos( ) 0 0
[ ] 0 cos( ) 0
0 0 sin( )
(1, ) , (2, ) , (3, ) 1,2,...,8
(1, ) , (2, ) 1, (3, ) , (4, ) , (5, )
(6, ) , (7, ) , (8, )
i i i
i
i i i
x R x R x R
i i
l Rii i i
l R l R l Rii
n
T n
n
R i e R i e R i e i
A i A i A i A i A i eR
A i e A i e A i e iR
1,2,...8
- Matrix 3 8fR
is defined by;
0...0
...
0...0
][)/()/( 81 RxiRxi
f eeR
94
- Expressions L and NL for cylindrical shells are given by:
2
2
2
2 2
2
2
1
1
1
2 3 1
2 2
L
U
x
VW
R
V U
x R
W
x
W V
R
W V U
R x R x R
and
2 2
2 2
2
1 1 1
2 8
1 1 1
2 8
1
2
0
0
0
NL
W V U
x x R
W V UV
R x R
W W WV
R x x
- Matrices pqB and pqC
/p q x R
pq pqB b e
/p q x R
pq pqC c e
95
where;
1 1 2
2
1 1sin
2 4pq pq pqb a b n
R
1 2
2
1sin cos
4pq pq pqc c c n n
R
with;
1
pq p p p q q qa n n
1
pq p qb n n , 1,...,8p q
1
pq p qc n
- Matrices for Equations (46-47)
T T
f f
pD R T T R dA
a
T Tk
f f
iG p M R T T R dA
R
3
33
3 3
1
12
T Tkfnl f
iG M R T T R dA
R
33
3
3 3
11
4
T Tkfnl f
iMH R T T R dA
a R
3
3 2
12
4
T Tkfnl f
iMH R T T R dA
a R
3
3 3
1 1
12
T T
fnl fD R T T R dAa
96
4.11 References
[1] H. Ashley and G. Zartarian, Piston Theory-New Aerodynamic Tool for Aeroelastician, Journal
of the Aeronautical Sciences, 23 (1956),1109-1118.
[2] W. Horn et al, Recent Contributions to Experiments on Cylindrical Shell Panel Flutter, AIAA
Journal, 12 (1974), 1481-1490.
[3] M.D. Olson and Y.C. Fung, Supersonic Flutter of Circular Cylindrical Shells Subjected to
Internal Pressure and Axial Compression, AIAA Journal, 4(1966), 858-864.
[4] M.D. Olson and Y.C. Fung, Comparing Theory and Experiment for Supersonic Flutter of
Circular Cylindrical Shells, AIAA Journal, 5(1967), 1849-1856.
[5] D.A. Evensen and M.D. Olson, Nonlinear Flutter of a Circular Cylindrical Shell in
Supersonic Flow, NASA TND - 4265, 1967.
[6] E.H. Dowell, Flutter of Infinitely Long Plates and Shells, Part I, AIAA Journal, 4(1966),
1510-1518.
[7] E.H. Dowell, Aeroelasticity of Plates and Shells, Noordhoff International Publishing, Leyden,
1975.
[8] G.W. Barr and R.O. Stearman, Aeroelastic Stability Characteristics of Cylindrical Shells
Considering Imperfections and Edge Constraint, AIAA Journal, 7(1969), 912-919.
[9] G.W, Barr and R.O. Stearman, Influence of a Supersonic Flow-field on the Elastic Stability of
Cylindrical Shells, AIAA Journal, 8(1970), 993-1000.
[10] M. Amabili, and F. Pellicano, Nonlinear Supersonic Flutter of Circular Cylindrical Shells,
AIAA Journal, 39(2001), 564-573.
[11] M. Amabili and F. Pellicano, Multimode Approach to Nonlinear Supersonic Flutter of
Imperfect Circular Cylindrical Shells, Journal of Applied Mechanics, Transactions ASME,
69(1976), 423-435.
[12] A.A. Lakis and M.P. Paidoussis, Dynamic Analysis of Axially Non-Uniform Thin
Cylindrical Shells, Journal of Mechanical Engineering Science, 14(1972), 49-71.
[13] M. N. Bismarck-Nasr, Finite Element Method Applied to the Supersonic Flutter of Circular
Cylindrical Shells, International Journal for Numerical Methods in Engineering, 10(1976), 423-
435.
97
[14] D.A. Evensen, Nonlinear flexural vibrations of thin-walled circular cylinders, NASA
Technical Note, Langley Research Center, NASA- TN D-4090,1967.
[15] D.A. Evensen, Nonlinear vibrations of an infinitely long cylindrical shell. American Institute
of Aeronautics and Astronautics Journal. 6(1968), 1401-1403.
[16] F. Sabri and A.A. Lakis, Finite Element Method Applied to Supersonic Flutter of Circular
Cylindrical Shells, AIAA Journal, 48 (2010), 73-81.
[17] A. Selmane, and A.A. lakis, Non-Linear Dynamic Analysis of orthotropic Open Cylindrical
Shells Subjected to a Flowing Fluid, Journal of Sound and Vibration, 202(1997), 67-93.
[18] A.A. Lakis and A. Laveau, Non-Linear Dynamic Analysis of Anisotropic Cylindrical Shells
containing a Flowing Fluid, International Journal of Solids and Structures, 28(1991),1079-1094.
[19] J.L. Sanders, An Improved First-Approximation Theory for Thin Shell, NASA R-24, 1959.
[20] M.H. Toorani, and A.A. Lakis, General equations of Anisotropic Plates and Shells Including
Transverse shear Deformations, Rotary Inertia and Initial Curvature Effects, Journal of Sound
and Vibration, 237(2000), 561-651.
[21] H.R. Radwan and J. Genin, Non-linear vibrations of thin cylinders. Journal of Applied
Mechanics, 43(1976), 370-372.
[22] H. Krumhaar, The Accuracy of Linear Piston Theory When Applied to Cylindrical Shells,
AIAA Journal, 1(1963), 1448-1449.
[23] R. Rougui and F. Moussaoui and R. Benamar, Geometrically non- linear free and forced
vibrations of simply supported circular cylindrical shells: A semi-analytical approach,
International Journal of Non-linear Mechanics. 42(2007), 1102-1115.
[24] M. Amabili, F. Pellicano, M.P. Paidoussis, Non-linear vibrations of simply supported circular
cylindrical shells coupled to quiescent fluid, J. Fluids Struct. 12(1998), 883-918
[25] M. Ganapathi and T.K. Varadan, Large amplitude vibrations of circular cylindrical shells,
Journal of sound and vibration, 192(1996), 1-14.
98
CHAPITRE 5 : DISCUSSION GÉNÉRALE
Cette étude représente l’analyse de flottement et la vibration non linéaire des coques cylindriques
circulaires vides et soumises à un écoulement supersonique. L’étude des différentes coques
analysées dans la littérature, aux nombres d’onde circonférentielle faible ou élevée, nous a permis
de savoir exactement le comportement complet de ces structures et de clarifier le désaccord entre
les chercheurs dans ce domaine d’aéroélasticité. Plusieurs paramètres ont été analysés à
différentes conditions limites et modes axiaux. Dans tous les cas, on a conclu que le degré de la
non-linéarité dépend de l’épaisseur, du rayon, de la longueur et des modes circonférentiels et
axiaux considérés. Le type de non-linéarité (Softening ou Hardening) est déterminé par un
paramètre nommé le rapport d’aspect.
Dans le cas où les coques cylindriques circulaires sont soumises à l’écoulement supersonique, le
modèle développé se consacre à la combinaison de la non-linéarité des coques (grandes
amplitudes de vibration des parois) analysées largement dans le premier article et l’écoulement
supersonique externe sous sa forme linéaire. Les résultats numériques fournissent le
comportement linéaire de ce type de panneaux pour différentes conditions limites et prédisent le
début du flottement et aussi le comportement non linéaire au-delà du début du flottement
caractéristique qui ne peut être prélevé par une analyse linéaire. Plusieurs paramètres dépendant
de la géométrie de la coque et de l’écoulement du fluide sont analysés. La non-linéarité la plus
dominante est de type softening et une faible instabilité de flottement est détectée, l’instabilité
dynamique est produite sous forme de flottement de modes couplés. La pression interne entraine
une stabilisation des coques et retarde le flottement en comparaison des cas de coques soumises à
aucune pression interne. Dans la dernière partie, on a essayé d’analyser l’effet des non-linéarités
de la structure et de l’écoulement supersonique du fluide afin de présenter une étude complète et
profonde non linéaire de ce type de structure. Le même type de non-linéarité, effet stabilisant de
la pression interne, et l’instabilité de flottement légère ont été remarqués sur tous les résultats
trouvés, aucune différence dans la variation des paramètres testés n’a été observée avec la
considération de la non-linéarité de l’écoulement supersonique du fluide exprimée par l’équation
de la théorie de Piston de troisième ordre. Ceci nous permet de conclure que l’effet d’une telle
non-linéarité n’a aucune influence sur les calculs et on peut la négliger devant la non-linéarité
géométrique (grande amplitude de vibration), ce qui est en bon accord avec les moindres
publications dans ce sujet.
99
CONCLUSIONS ET RECOMMANDATIONS
Cette recherche nous permet de développer un code efficace basé sur les éléments finis hybrides
traitant l’analyse linéaire et non linéaire d’aéroélasticité des coques cylindriques. Ce logiciel peut
être utilisé efficacement pour le design avancé de structures aérospatiales et la prédiction du
comportement des différents stages de flottement des coques soumises à l’écoulement
supersonique pour différentes conditions limites complexes et différents paramètres de la
géométrie et de fluide.
Ce logiciel développé permet :
- la modélisation linéaire et non linéaire des coques cylindriques par la méthode des
éléments finis hybride;
- la modélisation de l’effet de contraintes créé par la pression interne et la compression
axiale;
- la modélisation de la pression aérodynamique externe supersonique linéaire et non
linéaire donnée par la théorie de Piston
et est capable :
- de faire l’analyse dynamique linéaire et non linéaire des coques vides et soumises à
l’écoulement supersonique;
- de prédire le début du flottement et le comportement après le début de flottement;
- effet des contraintes initiales sur la stabilité linéaire et non linéaire aéroélastique.
Ce logiciel, résultat d’un travail analytique-numérique performant, précis et moins coûteux au
sens du temps de calcul, est considéré plus efficace que des programmes couplés en MEF et
CFD.
Les futurs travaux seront focalisés sur l’étude de l’effet de la température, qui est sûrement un
paramètre très important au niveau du design des structures aérospatiales et qui entre dans la
discipline d’ingénierie d’aérothermoélasticité. Les structures d’aujourd’hui font face à la
température élevée qui peut influencer la stabilité de ces structures à travers la distribution de la
100
température dans l’épaisseur des parois et aussi sur l’étude des différentes géométries des
panneaux utilisés dans l’industrie aérospatiale : les coniques, les plaques courbées,…afin
d’aboutir à la modélisation et l’analyse aérothermoélastique d’un véhicule complet, soit dans le
domaine fréquentiel ou temporel, soumise à un écoulement supersonique.
101
BIBLIOGRAPHIE
Bishlinghoff, R. L., Ashley, H. and Halfman R. L. (1955). Aeroelasticity, Addison-Wesley
publishing Company, Inc.
Dowell, E. H. (1975). Aeroelasticity of Plates and Shells, Noordhoff, International Publishing,
Lyden, The Netherlands.
Librescu, L. (1975). Elastostatics and Kinetics of anisotropic and Heterogeneous Shell-type
Structures. Noordhoff International Publishing, Lyden.
Fung, Y. C. (1960). A summary of the theories and Experiments on Panel Flutter. AFSOR TN
60-224. Guggenheim Aeronautical Lab. California Institute of technology, Pasadena, CA.
Johns, D. J. (1961). The Present Status of Panel Flutter. AGARD Report 484. Advisory Group
for Aeronautic Research and Development
Johns, D. J. (1965). A Survey on Panel Flutter, Advisory report 1, Advisory Group for
Aeronautic& Research and Development
Dowell, E. H. (1970). Panel Flutter: A Review of the Aeroelastic stability of Plates and Shells.
AIAA Journal, 8(3), 385-399.
Mei, C., Abdel-Motagaly, K., and Chen, R. (1999). Review of Nonlinear panel flutter at
Supersonic and Hypersonic speeds. Applied Mechanics reviews, 52(10), 321-332.
Gray, C. E. And Mei, C. (1993). large Amplitude Finite element Flutter analysis of Composites
panels in hypersonic Flow. AIAA Journal, 31(6), 1090-1099.
McIntosh, S. C., Reed, R. E. and Rodden, W. P. (1981). Experimental and theoretical study of
nonlinear flutter. Journal of Aircraft, 18, 1057-1063.
Reed, W. H. (1967). Review of Propeller-Rotor Whirl Flutter, NASA TR R-264.
Bismarck-Naser. (1978).Finite element method applied to the supersonic flutter of circular
cylindrical shells. International journal for numerical method in engineering, 10( 4), 423-435.
Bismarck-Naser. (1977). Finite element method applied to the flutter of two parallel elastically
coupled flat plates. International journal for numerical method in engineering, 11(7), 1977,1188-
1193.
102
Zhou R C, Xue D Y, Mei C. (1994). A finite element time domain-modal formulation for
nonlinear flutter of composite panels. AIAA J, 32(10), 2044–2052.
Hedgepeth, J. and Widmayer, E. (1963). Dynamic and Aeroelastic Problems of Lifting Re-Entry
Bodies. Aerospace Engineering, pp.148-153.
Dugundji, J. (1958). Effect of Quasi-Steady Air Forces on Incompressible Bending–Torsion
Flutter, Journal of the Aeronautical Sciences, 25( 2), 119–121
Cunningham, H.J., Batina, J .T., and Yang, H.T. (1991). Aeroelastic Analysis of Wings Using the
Euler Equations with a Deforming Mesh. Journal of Aircraft, 28, 778-788.
Timoshenko, S., Woinowsky-Krieger, S. (1959).Theory of Plates and Shells, New York:
McGraw-Hill.
Eisley, J. G. (1956). The flutter of a Two- Dimensional Buckled Plate with Clamped edges in a
supersonic flow, OSR-TN-56-296.
Houboult, J. C. (1958). A Study of several Aerothermoelastic problems of Aircraft structures in
High-Speed Flight. Ph.D Thesis. The Swiss Federal Institute of Technology. Zurich:
Eidenossischen technischen Hochschule.
Fung, Y.C. (1958). On Two-Dimensional panel flutter. Journal of the Aeronautical sciences, 25,
145-160.
Bolotin, V. V. (1963). Nonconservative Problems of the Theory of Elastic Stability. New York:
McMillan Co, pp. 199-312.
Chu, H.N. And Herrmann, G. (1956). Influence of Large Amplitudes on Free Flexural Vibration
of rectangular elastic Plates. Journal of applied Mechanics, 23, 532-540.
Ashley, H. And Zaratarian, G. (1965). Piston Theory - A New Aerodynamic Tool for the
Aeroelastician. Journal of Aeronautical science, 23(10), 1109-1118.
Dowell, E. H., and Voss, H. M. (1965). Experimental and theoretical panel flutter Studies in
mach Number range 1.0 to 5.0. AIAA Journal, 3(12), 2292-2304.
Dowell, E, H. (1966). Nonlinear Oscillation of a fluttering Plate I. AIAA Journal, 4(7), 1267-
1275.
103
Dowell, E. H. (1967). Nonlinear Oscillation of a Fluttering Plate II. AIAA Journal, 5(10), 1856-
1862.
Fralich, R. W. (1965). Postbuckling effects on the flutter of simply supported rectangular panels
at Supersonic speed. NASA TN D-1615.
Ventres, C.S. (1970). Nonlinear flutter of Clamped Plates. Ph.D, Dissertation. Princeton
University.
Librescu, L. (1965). Aeroelastic stability of orthotropic heterogeneous thin Plates in the Vicinity
of flutter critical Boundary. Journal de Mecanique, 4(1), 51-76.
Kobayashi, S. (1962). Flutter of Simply Supported Rectangular Panels in a Supersonic Flow-
Two- Dimensional Panel Flutter, I- Simply supported panel, II- Clamped Panel. Transaction of
Japan Society of Aeronautical and Space Sciences, 5, 79-118.
Yuen, S. W., and Lau, S. L. (1991). Effect of in- Plane load on nonlinear Panel Flutter by
Incremental Harmonic Balance Method. AIAA Journal, 29(9), 1472-1479.
Eslami, H. (1978). Nonlinear flutter and Forced Oscillations of Rectangular Symmetric Cross-Ply
and Orthotropic Panels using harmonic balance and Perturbation Methods, Ph.D. Dissertation.
Old Dominion University.
Morino, L. (1969). A Perturbation Method for Treating Nonlinear Panel Flutter problems. AIAA
Journal, 7(3), 405-410.
Morino, L, and Kuo, C. C. (1971). Detailed extensions of perturbation Methods for Nonlinear
Panel flutter. ASRL TR 164-2, M.I.T. Cambridge.
Kuo, C. C., Morino, L., and Dugundji, L. (1972). Perturbation and harmonic balance for Treating
Nonlinear Panel Flutter. AIAA Journal, 10(11), 1479-1484.
Olson, M. D. (1967). Finite element Approach to Panel Flutter. AIAA Journal, 5(12), 226-227.
Appa, K. and Somashekar, B. R. (1969). Application of Matrix Displacement Methods in the
study of Panel Flutter. AIAA Journal, 7(1), 50-53.
Sander, G., Bon, C. and Geradin, M. (1973). Finite element Analysis of supersonic Panel Flutter,
International Journal for Numerical Methods in Engineering, 7(3), 379-394.
104
Yang, t. Y. and Sung, S H. (1977). Finite element in three- Dimensional supersonic Unsteady
Potential Flow. AIAA Journal, 15(12), 1677-1683.
Mei, C. (1977). Finite Element in Approach for Nonlinear Panel Flutter. AIAA Journal, 15(8),
1107-1110.
Mei, C. and Rogers, J. L. (1976). Application of Nastran to large Deflexion supersonic Flutter of
Panel. NASA TM X-3428, 67-98.
Rao, K. S. and Rao, G. V. (1980). Supersonic Flutter of Panels Elastically Restrained Against
Rotation. Computer and Structures, 11, 197-201.
Mei, C. and Weidman, D. J. (1977). Nonlinear Panel Flutter – A finite element Approach,
Computational Method for Fluid-Structure Interaction Problems, T.Belytsechke and T.L.Geers,
ASME, New York, 26, 944-951.
Mei, C. and Wang, H. C. (1982). Finite Element Analysis of Large Amplitude Supersonic Flutter
of Panels. Proceedings of international conference on Finite Element Methods. Shanghai, China,
Gordon and Breach Science Publishers, Inc. 944-251.
Han, A. D. and Yang, T. Y. (1983). Nonlinear Panel Flutter using High Order Triangular Finite
Element. AIAA Journal, 21(10), 1453-1461.
Green, P. D. and Killey, A. (1997), Time Domaine Dynamic Finite Element Modelling in
Acoustic Fatigue Design, in Proc., 6th
International Conference on Structural Dynamic, ISVR, 11,
1007-1026.
Robinson, J. H. (1990). Finite Element Formulation and Numerical Simulation of the large
deflection Random Vibration of Laminated composite Plates. MS Thesis. Old Dominion
University.
Chandiramani, N. K. and Plaut, R. H, and Librescu, L. L. (1995). Nonperiodic Flutter of a
Buckled Shear deformable Composite Panel in a high-supersonic Flow. International Journal of
Non-Linear Mechanics. 30(2), 149-167.
![Linear Global Translation Estimation with Feature …Recently, Wilson and Snavely [46] directly solved cameras and points by Ceres Solver [1] after applying a smart filter to essential](https://img.pdfslide.fr/doc/110x75/5f805e6b6050b07370169aec/linear-global-translation-estimation-with-feature-recently-wilson-and-snavely-46.jpg)
